AIIF
Algebra II Foundations
Non– Linear
Functions
Teacher Manual
Table of Contents
Lesson
Page
Lesson 1: Introduction to Quadratic Functions................................................................................................1
Lesson 2: The Quadratic Formula ....................................................................................................................17
Lesson 3: Graphing Quadratic Functions and Their Applications .............................................................35
Lesson 4: Power Functions................................................................................................................................63
Lesson 5: Inverse Variation...............................................................................................................................89
Lesson 6: Exponential Functions....................................................................................................................107
Lesson 7: Step Functions .................................................................................................................................135
Lesson 8: Miscellaneous Non–Linear Functions..........................................................................................158
Assessments ......................................................................................................................................................193
CREDITS
Author:
Contributors:
Graphic Design:
Dennis Goyette and Danny Jones
Robert Balfanz, Dorothy Barry, Leonard Bequiraj, Stan Bogart, Robert Bosco, Carlos Burke, Lorenzo
Hayward, Vicki Hill, Winnie Horan, Donald Johnson, Kay Johnson, Karen Kelleher, Kwan Lange, Dennis
Leahy, Song-Yi Lee, Hsin-Jung Lin, Guy Lucas, Ira Lunsk, Sandra McLean, Hemant Mishra, Glenn Moore,
Linda Muskauski, Tracy Morrison, Jennifer Prescott, Gerald Porter, Steve Rigefsky, Ken Rucker, Stephanie
Sawyer, Dawne Spangler, Fred Vincent, Maria Waltemeyer, Teddy Wieland
Gregg M. Howell
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All rights reserved. Student assessments, Cutout objects, and transparencies may be duplicated for classroom use only; the number
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content, IP addresses changes, pop advertisements, or redirects. It is further recommended that teachers confirm the validity of the
listed addresses if they intend to share any address with students.
AIIF
Non–Linear Functions
Planning Document
Page i
Planning Document: Non-Linear Functions
Overview
The types of non-linear functions include:
• Quadratic functions
• Power functions
• Inverse variation functions
• Exponential functions
• Step functions
• Absolute value functions
• Circles (domain restricted to be a function)
• Piece-wise functions
The number of total suggested days for the unit is 18. Adjustments may be needed based on student
performance during the unit and amount of time available until the end of the semester.
Vocabulary
Square/squaring
Quadratic equation
Square root
Minimum point
Maximum point
Standard form of a quadratic function
General form of a quadratic function
Parabola
Solutions
Quadratic formula
Discriminant
Double root
Break-even point
Symmetry
Vertical line symmetry
Vertex
x-coordinate
y-coordinate
Quadratic regression
Multiples
Power function
Even function
Odd function
Direct variation
Inverse variation
Constant of proportionality
Base
Exponent
Exponential function
Growth
Decay
Exponential regression
Rise
Run
int() function
Greatest integer
Floor function
Smallest integer
Ceiling function
Binary number system
Absolute value function
Dilation
Vertical line test
General equation of a circle
Piece-wise function
Material List
Student journal
Setting the Stage
transparencies
Dry-erase boards
Markers and erasers
Chart paper
Graphing calculators
Calculator view screen
Blank transparencies
Lesson specific
transparencies
Overhead projector
Construction paper
Poster paper
Colored pencils
1 Day
2 Days
3 Days
2 Days
2 Days
3 Days
The Quadratic
Formula
Graphing Quadratic
Functions and Their
Applications
Power Functions
Inverse Variation
Exponential Functions
Timeline
Introduction to
Quadratic Functions
Lesson
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Write and solve simple quadratic equations
Use the graphing calculator to find vertex, xintercepts, and to draw a graph
Standard form of a quadratic function
Quadratic formula
Discriminant
Solve quadratic functions using the quadratic
formula
y-intercept
Applications of quadratic functions
Line of symmetry
Vertex
Graphing
Applications of quadratic functions
Quadratic regression
Power function format
Odd and even functions
Graphing power functions
Transformations involving power functions
Applications of power functions
Write equations involving direct variation
Constant of proportionality
Write equations involving inverse (indirect)
variation
Graph direct and inverse variation
Identify inverse variation phrases
Identify exponential functions
Growth and decay
Exponential applications
Graph exponential functions
Concepts Covered
How does inverse variation affect real–
world application problems?
How does direction variation affect
real–world application problems?
How do exponential functions behave in
real-world applications?
•
•
Do power functions have patterns that can
be used when solving and graphing them?
How can quadratic functions and
applications of quadratic functions be
graphed?
How can the quadratic formula be used to
solve real-world applications?
How does the process of squaring relate to
quadratic functions?
Essential Question(s)
The following table contains lesson name, timeline, summary of concepts covered, and the Essential Question(s) for each lesson.
Page ii
AIIF
2 Days
3 Days
Step Functions
Miscellaneous NonLinear Functions
Non–Linear Functions
Planning Document
•
•
•
•
•
•
•
•
•
Rise and run of a step function
Floor step function
Ceiling step function
Graph step functions using the graphing
calculator and the int() function
Applications involving step functions
Absolute value functions
o Transformations involving absolute value
functions
o Graph absolute value functions
Circles
o Equations for circles
o Restrict domain of a circle equation
o Solve circle equations for y
o Graph circle equations using center,
radius, and intercepts
Piece-wise functions
o Write piece-wise functions
o Graph piece-wise functions
o Applications of piece-wise functions
How do absolute value functions, piece–
wise functions, and circle equations apply
to real–world applications?
How do step functions apply to real-world
applications?
Page iii
AIIF
Non–Linear Functions
Lesson 1: Introduction to Quadratic Functions
AIIF
Page 1
Lesson 1: Introduction to Quadratic Functions
Objectives
• Students will be able to identify quadratic functions
• Students will be able to use the squaring process to create quadratic functions
• Students will be able to solve simple quadratic functions by taking the square root
• Students will be able to graph quadratic functions
• Students will be able to determine the maximum or minimum of a quadratic function
Essential Questions
• How does the process of squaring relate to quadratic functions?
Tools
• Student Journal
• Setting the Stage transparency
• Dry–erase boards, markers, erasers
• Chart paper
• Graphing calculators
Warm Up
• Problems of the Day
Number of Days
• 1 Day
Vocabulary
Quadratic equation
Maximum point
Horizontal translation
Square/Squaring
Minimum point
Square root
Vertical translation
Notes
Prior to teaching, you will need to prepare transparencies from the master hard copies supplied in this
manual.
•
At the end of each lesson in Algebra II Foundations there are Practice Exercises, Outcome Sentences, and a
small quiz. The authors suggest that teachers use these tools as needed and as time allows.
•
AIIF
Page 2
Teacher Reference
Setting the Stage
Place the Setting the Stage transparency on the overhead projector but cover it at first. Give each student a
piece of string 10 inches long (the grid squares on the dry–erase boards are 5/16 of an inch square). Eight
squares make 2.5 inches. Have the students make a square from their piece of string on their dry–erase board
(grid side) that is 8 units by 8 units. After the class has made a square, 8 X 8, uncover the transparency. Ask the
class the following questions or something similar:
•
•
•
•
•
•
•
What are the lengths of the sides of your square?
What is the area of your square?
What is the formula for the area of a square?
Write an equation that represents the area of your square (we are looking so see if the students come up
with something like x2 = 64 or s2 = 64. Some students might write A = 64 square units.) The key is that the
students see that we need to square the length of a side of the square to get the area.
What is the exponent of your variable in your equation?
What is the term we use when we raise something to the second power? (We are looking for students to
use the term square or squaring.)
What is the opposite operation called? (We are looking for students to use the term square root.)
Tell the class that equations/functions that have the independent variable to the second power, or squared, are
called quadratic equations or quadratic functions respectively. You might want to tell the class that the term
quadratic comes from the Latin word quadratus, which means square. Ask the class how the Latin word
quadratus is related to the term quadratic. The key concept is that quadratic has a variable term with the highest
exponent being 2, meaning squared. Some Latin words you may want to use are quadratum for square, or
quad meaning 4. You could ask the class to name mathematic terms that start with the word quad like
quadrilateral, quadrants, quadrangle, and quadruple.
Non–Linear Functions
Lesson 1: Introduction to Quadratic Functions
AIIF
Page 3
Setting the Stage
Transparency
AIIF
Page 4
Teacher Reference
Activity 1
In this activity, students will create, work with, and solve simple quadratic equations. Think aloud as you
model the following problem:
The area of a square yard is 49 square meters. What are the dimensions of the yard? What is the perimeter of the yard?
Even though some students might be able to give the answer right away, make sure to model how to properly
set up and solve these types of equations. You could refer to the table method in the Solving Equations unit.
Some of the concepts you should model are:
• Label variable(s) (here we would label or state that the length equals the width which we could label as
s for the length of the sides of the square yard.)
• Write an equation, using your variable(s) representing the problem (here, using our variable we would
have s 2 = 49 .)
• Solve the equation, make sure the answer contains appropriate units (the solution is s = 7 meters.)
Have a volunteer model the following problem, on the overhead projector, while the class follows along on
their dry-erase boards: A number squared is 36.
If the students come up with only one solution, ask, "Is there any other number that when squared equals 36?"
The key is for the students to understand that both a negative six and a positive six, when squared, would
equal 36. Now ask the class, "Why was there only one answer for the unknown in the problem related to the
dimensions of the yard with an area of 49 square meters?" The goal is for students to differentiate between
problems with just numbers and variables and real–world applications. Sometimes answers just don’t make
sense in the real world even though mathematically they are correct. For example, you cannot measure using
negative numbers.
Have another volunteer model the following problem, on the overhead projector, while the class follows
along, using dry–erase boards: The square of the sum of a number and three is sixteen.
This problem is a bit more complicated than the previous one. For the class to write the correct equation you
may want to ask guiding questions such as “What is being squared?”, "What letter should be used for the
variable", and “What must be calculated first, the sum or the square?” These questions are to help the students
come up with the equation ( x + 3)2 = 16 . The questions are also to determine how the students will handle
solving a quadratic equation. You will want to let the students explore different strategies to determine how to
solve this equation and then talk about their strategies. The students can refer back to the table method.
Have the students work individually on exercises 1 through 6. After the students have completed the exercises
have them pair up with another student to go through the exercises together and verify answers. If they
disagree on any of the answers, have them check with another student pair or pairs until they agree. Have
volunteers share their answers with the class on any exercises not agreed upon. Circulate while the students
are working to ask guiding questions and encourage the students. Note: exercises 5 and 6 may need some
additional explanation.
AIIF
Non–Linear Functions
Lesson 1: Introduction to Quadratic Functions
Page 5
Activity 1
SJ Page 1
For the following exercises, write a matching quadratic equation then solve the equation and answer each
question.
1.
The square of a number is sixty-four. What are the numbers that make the equation true?
The quadratic equation is n 2 = 64 . Solving this equation we get that the number can be 8 or –8.
2.
The square of the difference of a number and six is one hundred twenty-one. What are the numbers?
The quadratic equation is ( n − 6 ) = 121 . The answers are n = –5 and n = 17.
2
3.
4.
5.
6.
The area of a square piece of paper is 144 square inches. What are the lengths of the sides of the square
piece of paper? The quadratic equation is n 2 = 144 . The lengths
Crop circles are patterns
of the sides of the square are 12 inches.
created by the flattening of
crops
such as wheat, barley,
The area of a crop circle is approximately 1257 square feet.
rapeseed, rye, corn, linseed,
Approximately what is the size of the diameter of the circle? Use
and soy into circles. The term
2
3.14 for π . The quadratic equation is π r = 1257 . Solving this
was first used by researcher
equation we get r ≈ 20. Because the radius is approximately 20
Colin Andrews to describe
feet, this makes the diameter of the crop circle about 40 feet.
simple circles he was
researching.
A company’s cost can be determined by the sum of the variable
costs and the fixed costs. Namely, C(x) = variable costs + fixed
costs. It has been determined that the company’s variable costs are the cost of producing one unit times
the square of the number of units.
a.
If it costs $11.00 to produce a single unit of the product and the company's fixed costs are
$1,000.00, write an equation, in function notation, that represents the total costs of producing x
units of the product. The quadratic cost function is C( x ) = 11x 2 + 1000 .
b.
If total costs are $2,751,000.00, how many units of the product are produced?
There are 500 units of the product produced.
A company makes a product. The company has determined the approximate cost to produce a single
unit of the product. The company has fixed costs of $500. The company also knows that it costs
$250,500 to produce 100 units of the product. The engineering department’s research shows that the
variable portion of the cost function is the cost to produce a single unit times the square of the number
of units produced. That is, variable costs = cx 2 . Write a cost function, C(x), which represents the total
cost of producing x units of the product. Use the information given to determine the cost to produce
one unit of the product. The quadratic equation is C( x ) = cx 2 + 500 . The cost per unit of the product is
$25.
250500 = c100 2 + 500
250500 − 500 = 10000c + 500 − 500
250000 = 10000c
250000
= 25 = c
10000
AIIF
Page 6
Teacher Reference
Activity 2
In this activity, students will investigate the graphs of quadratic functions using the classroom graphing
calculator. Use the Parallel Modeling strategy to model graphing y = x2 while students model the same or a
similar equation. For this first equation, have the students model the same equation with you. You may want
to prepare a list of questions to ask the students after the graph is displayed on the graphing calculator view
screen and on the students' graphing calculators. Here are some questions that you could ask:
• What is the shape of the graph?
• What do you notice about the portion of the graph to the left of the y-axis compared to the portion to
the right of the y axis?
• What do you know about minimum and maximum values?
• Does the graph have a minimum or maximum value?
Have a volunteer record the responses and characteristics on the board while the students record the responses
and characteristics in their student journals.
Next, have a volunteer model graphing the equation y = –x2 in the front of the class. Have a second volunteer,
along with the other students, list the similarities and differences between this equation and y = x2. The goal is
that the students notice that the graphs are exactly the same except that y = –x2 is “flipped”, or reflected, about
the x–axis and is the opposite of y = x2.
Now have another volunteer model the equation y = 2x2 while the class models y = 3x2. Have a second
volunteer record the class' responses about the similarities between the very first equation graphed, y = x2, and
the equation they have graphed now. You may want students to display both graphs at the same time to make
the comparison easier.
Have the class get back in their pairs from Activity 1. Give each pair a piece of chart paper. Have the pairs
complete a similar comparison for y = –2x2 and y = –3x2. You may decide to have the students display all 3
graphs on their calculators at the same time (y = –x2, y = 3x2, y = –3x2, and maybe even y = x2). Have the pairs
list the similarities and differences between the graphs on their chart paper and draw a copy of the graphs
from the graphing calculator to the chart paper. Give the students five to seven minutes to complete the task
and then have the class display their chart paper along the walls of the classroom. The class can then walk
around checking other pair's graphs and list of similarities and differences. Lead a class discussion on the
similarities and differences that the class listed. You may want to show the class how to use the graphing
calculator capabilities to find the maximum or minimum y–value of each graph. The graphing calculator can
display the coordinates of the minimum or maximum value. The steps and screen shots below pertain
specifically to the TI-83 or 84 Plus™ graphing calculator.
Steps to Find the Maximum/Minimum
• Press the 2ND key followed by the TRACE key to display the CALCULATE menu (assuming the
equation has been entered into the Y= editor and the graph is displayed).
• Press 3 for minimum or 4 for maximum
• Use the arrow keys to move the blinking cursor to the left of the minimum or maximum point on the
graph (calculator is looking for the Left Bound) and press the ENTER key.
• Use the arrow keys to move the blinking cursor to the right of the minimum or maximum point on the
graph (calculator is looking for the Right Bound) and press the ENTER key.
Non–Linear Functions
Lesson 1: Introduction to Quadratic Functions
AIIF
Page 7
•
•
Use the arrow keys to move the blinking cursor to the minimum or maximum point on the graph
(calculator is looking for your Guess?) and press the ENTER key.
Minimum or maximum coordinates are displayed at the bottom of the screen
The screen shots below are for the above steps.
Mention the TRACE key to the students which will allow them to trace along the graphs. Also mention that
the up and down arrow keys will allow students to switch between graphs in order to distinguish which graph
represents which equation.
Model, or have a student model, graphing the equation y = x2 + 1 while the class parallels with y = x2 + 2. Ask
the students, "What have you noticed about the minimum value of y and its coordinates?" The students should
realize that the minimum value of y and its coordinates have been translated vertically by two units or one
unit, respectively. You could also model an equation where the vertical translation is negative so that the
students understand that the constant added will tell them what the minimum or maximum value is as well as
its coordinates.
Now model, or have a student model, graphing the equation y = (x – 2)2 while the class parallels with
y = (x – 1)2. Ask the students, "What have you noticed about the minimum value of y and its coordinates?" The
students should realize that the minimum value of y and its coordinates have been translated horizontally by
two units or one unit, respectively. You could also model an equation where the horizontal translation is
negative so that the students understand that the constant added with x before it is squared will tell them the
x–coordinate of the minimum value.
Now model, or have a student model, graphing the equation y = (x – 2)2 + 2 while the class parallels with
y = (x – 1)2 + 1. Ask the students, "What have you noticed about the minimum value of y and its coordinates?"
The students should realize that the minimum value of y and its coordinates have been translated both
vertically and horizontally. You could also model equations where the horizontal and vertical translations are
a combination of positive and negative values.
AIIF
Page 8
As a last modeling example, display the graph of the equation y = (x +1)2 – 4. Model to the class how the
coordinates of the vertex can be used to determine the equation for the graph. From the previous examples, we
know that adding/subtracting a constant from x before squaring creates a horizontal translation while
adding/subtracting a constant from the squared term creates a vertical translation. Show the students how
using the coordinates of the vertex from the graph, they can obtain the equation y = (x +1)2 – 4. The graph is
displayed below.
Have the students work in pairs on Exercises 1 through 10. After the students have completed the exercises,
have each pair join another pair and compare their findings. Have the students include in their work
contributions given by the other student pair in their group. Bring the class together and have volunteers share
their results with the class.
Non–Linear Functions
Lesson 1: Introduction to Quadratic Functions
AIIF
Page 9
Activity 2
SJ Page 2
In this activity, you will investigate the graphs of various quadratic equations.
For the following exercises, find the coordinates of the minimum or maximum value and state the minimum or
maximum y–value, all x–intercepts and y–intercepts, and make a sketch of the graph in the grid provided. The
exercises are set up in most cases to draw two graphs per grid. You may also want to display the two graphs
simultaneously on your graphing calculator as well. NOTE: Displaying table values on your graphing
calculator may help you to draw the graph.
1.
y=
1 2
x
2
Minimum y–value is 0 and the coordinates of
the minimum value is (0, 0). Only x–intercept
and y–intercept is (0, 0).
2.
y = 4x 2
Minimum y–value is 0 and the coordinates of
the minimum value is (0, 0). Only x–intercept
and y–intercept is (0, 0).
3.
y = −x 2 + 3
Maximum y–value is 3 and the coordinates of
the maximum value is (0, 3); y–intercept is (0,
3). There are two x–intercepts:
(
)
(
)
3 ,0 and − 3 ,0 , or approximately
(1.73,0) and (–1.73, 0). The students may be
familiar with the decimals rather than the
radicals.
4.
y = −x 2 − 2
Maximum y–value is –2 and the coordinates
of the maximum value is (0,–2); y-intercept is
(0, –2). There are no x–intercepts.
AIIF
Page 10
SJ Page 3
5.
y = ( x + 3)
2
Minimum y–value is 0 and the coordinates
of the minimum value is (0,0); y–intercept is
(0, 9). Only x–intercept is (–3, 0).
6.
y = (x − 2)
2
Minimum y–value is 0 and the coordinates
of the minimum value is (2, 0); y–intercept is
(0, 4). Only x–intercepts is (2, 0).
7.
y = (x − 2) + 3
2
Minimum y–value is 3 and the coordinates
of the minimum value is (2, 3); y–intercept
is (0, 7). There are no x–intercepts.
8.
y = (x − 2) − 3
2
Minimum y–value is –3 and the coordinates
of the minimum value is (2, –3); y–intercept
is (0, 1). There are two x–intercepts:
approximately (3.73, 0) and (0.27, 0).
Non–Linear Functions
Lesson 1: Introduction to Quadratic Functions
AIIF
Page 11
SJ Page 4
9.
For the given graph, identify the
coordinates of the minimum point, x–
intercepts, y–intercept, and equation
representing the graph.
The minimum point has the coordinates
(–2, –9); the equation of the graph is y =
(x +2)2 – 9; the x–intercepts are (–5, 0) and
(1, 0); the y–intercept is (0, –5).
10.
From Exercises 1 through 9, what conclusions and characteristics can you make about the graphs of
quadratic equations?
Answers will vary. Sample response might be: The larger the number in front of x, the more narrow the
graph; positive numbers in the front of x2 give a minimum value while negative numbers in the front of
x2 give a maximum value; adding or subtracting a number from the square term vertically translates
the graph that number of units up or down; adding and subtracting a number from the variable and
then squaring it translates the graph horizontally by that number of units to the left or right.
AIIF
Page 12
SJ Page 5
Practice Exercises
Solve each of the following.
1.
The square of a number is one hundred forty-four. Write a quadratic equation and then solve for the
unknown number(s).
The equation is n 2 = 144 ; the numbers are 12 and –12.
2.
The square of the sum of a number and nine is one hundred sixty-nine. Write a quadratic equation and
then solve for the unknown number(s).
The equation is ( n + 9 ) = 169 ; the numbers are 4 and –22.
2
3.
The Sparkling Diamonds jewelry store sold a diamond studded bracelet and made a profit of $196. The
profit is based on the cost of the necklace to the store. How much did the necklace cost the store if profit
C2
⎛ C ⎞
is determined by the equation P = ⎜
, where P is the profit and C is the cost of the item?
•
C
=
⎟
100
⎝ 100 ⎠
The necklace cost the store $140.
4.
Graph the quadratic equation y = −3x 2 + 12 on the grid supplied below. Label all intercepts and
determine the maximum or minimum point.
The y–intercept is (0, 12); the x–intercepts are (2, 0) and (–2, 0). The maximum point is (0, 12).
5.
Graph the quadratic equation y = ( x − 4 ) on the grid supplied below. Label all intercepts and
2
determine the maximum or minimum point.
The y–intercept is (0, 16); the x–intercept is (4, 0). The minimum point is (4, 0).
4.
5.
AIIF
Non–Linear Functions
Lesson 1: Introduction to Quadratic Functions
Page 13
SJ Page 6
6.
Graph the quadratic equation y = ( x − 7 ) − 16 . Label all
2
intercepts and determine the maximum or minimum
point.
The y–intercept is (0, 33); the x–intercepts are (3, 0) and
(11, 0). The minimum point is (7, –16).
7.
For the given graph, identify the minimum
point, x–intercepts, y–intercept, and equation
representing the graph.
The minimum point is (–3, –4), the x–intercepts
are (–1, 0) and (–5, 0), the y–intercept is (0, 5),
the equation of the graph is y = (x + 3)2 – 4.
8.
How does the process of squaring relate to quadratic functions?
Answers will vary. Sample response: Because the term quadratic comes from a Latin word meaning "to
square," squaring the x variable makes the function quadratic compared to something else like linear.
AIIF
Page 14
SJ Page 7
Outcome Sentences
To solve a quadratic equation
I know a quadratic equation will have a minimum when
I know a quadratic equation will have a maximum when
The minimum or maximum of a quadratic equation can be determined by
I can use the graphing calculator to
I would like to find out more about
I now understand
I still have a question about
Non–Linear Functions
Lesson 1: Introduction to Quadratic Functions
AIIF
Page 15
Teacher Reference
Lesson 1 Quiz Answers
1.
y-intercepts (0, –32); the x-intercepts are (4, 0) and (–8, 0). The minimum point is (–2, –36).
2.
The equation is ( n − 6 ) = 100 . The unknown numbers are 16 and –4.
3a.
3b.
5 inches
6 inches
2
AIIF
Page 16
Lesson 1 Quiz
1.
Name:
Graph the quadratic equation y = ( x + 2 ) − 36 . Label all intercepts and determine the maximum or
2
minimum point.
2.
The square of the difference of a number and six is one hundred. Write a quadratic equation and then
solve for the unknown number(s).
3.
Oatmeal is a great nutritious breakfast on a cold morning. Most oatmeal containers are cylindrical. The
volume of a cylindrical container is given by V = π r 2 h , where r is the radius of the container, and h is
the height of the container. Find the radius of an oatmeal container when:
a.
The volume is 745.75 cubic inches and the height is 9.5 inches. Approximate π with 3.14. Round
answer to the nearest quarter inch.
b.
The volume is 678 cubic inches and the height is 6 inches. Assume 3.14 for pi. Round answer to
the nearest quarter inch.
Non–Linear Functions
Lesson 2: The Quadratic Formula
AIIF
Page 17
Lesson 2: The Quadratic Formula
Objectives
• Students will be able to use technology to understand, simplify, and solve quadratic equations.
• Students will be able to write quadratic equations in general form in order to use the quadratic formula.
• Students will be able to use the discriminant to determine the nature of the roots for a quadratic equation.
• Students will be able to use the quadratic formula to solve real–world applications.
Essential Questions
• How can the quadratic formula be used to solve real-world applications?
Tools
• Student Journal
• Setting the Stage transparency
• Blank transparencies
• Overhead projector
• Dry–erase boards, markers, erasers
• Graphing calculators
Warm Up
• Problems of the Day
Number of Days
• 2 days
Vocabulary
Standard form of a quadratic function
General form of a quadratic functioin
Quadratic formula
Zeros
Parabola
Discriminant
Break-even point
Solution
Double root
x-intercept
AIIF
Page 18
Teacher Reference
Setting the Stage
Talk with students about how computer programmers and designers are often trying to determine a quicker
method for computers to work, so that as you use programs or play video games the lag time is as short as
possible. This is done in basically two ways. One way is to make the hardware in the computer, such as the
computer chips and circuits, faster by physically making them smaller and with better conducting material so
that the electronic information can travel quickly within the computer. Another method is to design the
software (program) with the least amount of program language instructions as possible. Mathematics has a
major influence in both of these areas. The geometry, physics, and chemistry behind making the hardware
smaller is all built on math formulas. Often in the computer program, functions and equations are entered so
that the computer calculates and outputs a display as a number or colored pixel on the screen.
Display the Setting the Stage transparency 1. For example, a particular software program may need to display
a line or lines across the screen. In order to do this the program would need to calculate the distance between
two points. Which method do you think would save time and allow the program to run faster? Explain.
Display the second transparency when necessary to visually illustrate the two points and the line, representing
the distance, between the two points.
Talk with students that in math we often find methods to make steps shorter. In many cases you have learned
how to shorten the steps yourself. Today, you are going to encounter a short method to solve a quadratic
equation and skip many steps that would take weeks to teach you. By recognizing what values to input, you
will be able to calculate the output that solves the equation without using any solving techniques like you do
for linear equation. Enjoy the short cut!!!
AIIF
Non–Linear Functions
Lesson 2: The Quadratic Formula
Page 19
Setting the Stage
Transparency 1
Method 1
Step 1: Input x1, x2, y1, and y2
Step 2: Calculate y2–y1
Step 3: Square the value in Step 2
Step 4: Calculate x2–x1
Step 5: Square the value in Step 4
Step 6: Sum the values from Step 3 and Step 5
Step 7: Determine the square root of the value
in Step 6
Step 8: Output the value from Step 7
Method 2
Step 1: Input x1, x2, y1, and y2
Step 2: Calculate
( x 2 − x1 ) 2 + ( y 2 − y1 ) 2
Step 3: Output the value from Step 2
AIIF
Page 20
Setting the Stage
Transparency 2
(x2, y2)
(x1, y1)
Non–Linear Functions
Lesson 2: The Quadratic Formula
AIIF
Page 21
Teacher Reference
Activity 1
In this activity, students will write quadratic equations in standard and general form, and identify the
coefficients a, b, and c. Students will also be introduced to the quadratic formula and the importance of the
discriminant in determining the nature of the roots of the quadratic formula.
Introduce the class to the standard form of a quadratic equation: y = ax 2 + bx + c . Inform the class that the
coefficients a, b, and c will play a very important part in today’s lesson. Ask the students if they can determine
what the y–intercept will be for a quadratic equation written in standard form. We are gauging the student’s
prior knowledge of the y–intercept to determine if they remember that the x-value is always 0 for the y–
intercept.
Introduce the class to the general form of a quadratic equation: ax 2 + bx + c = 0 . Note: Many books and
resources have different ways to write the standard form and the general form. We will use the equations
stated above for our purposes. A quadratic function in standard form is written as f ( x ) = ax 2 + bx + c .
Ask the class what is the difference between the standard form and general form of a quadratic equation. The
students need to recognize that y is set equal to zero for the general form. If the class realizes that y = 0, then
ask the class “If y = 0 and we solve the quadratic equation in the general form for x, what do we get?” The
students should realize that we get the solutions and the x–intercepts of the quadratic equation. Another
name that they may not be aware of is called the zeros of the quadratic equation. They are called the zeros
when substituted for x; because, the quadratic equation evaluates to zero. Hence, solving a quadratic equation
gives solutions, x–intercepts, and the zeros. NOTE: The concept that we have three different names when
solving a quadratic equation can be expanded for all polynomials of degree higher than 2.
−b ± b 2 − 4 ac
, for solving quadratic equations (y =
2a
ax2 + bx +c) when y=0. Let the class know that this formula always works when trying to find the solution for a
quadratic equation. Point out to the class the importance of the portion under the radical sign, called the
discriminant (b2 – 4ac). Have the class get into groups of four. Ask the class “What are the three possible values
for the discriminant and how do they affect the solutions for x?” You might want to give the class a hint and
say that one of the possible values are positive numbers. Have the students work in their groups to discuss the
answer to the question. You might want to give the students 3 sets of values for a, b, and c to help them
determine that the three values we are looking for are positive, zero, and negative. Here are three sets of
values for a, b, and c:
Next, introduce the students to the quadratic formula, x =
Use: b2 – 4ac
1. a = 1; b = 2; c = –4
2. a = 1; b = 2; c = 1
3. a = 1; b = 2; c = 3
The class should discover the following:
1. When the discriminant is positive (greater than 0) then these are two roots that give two distinct
solutions for x.
2. When the discriminant is zero then there are two roots that are exactly the same that give a solution for
x, which we call a double root.
AIIF
Page 22
3. When the discriminant is negative (less than 0) then there are no real number solutions for x.
Bring the class back together and have groups share their findings with the rest of the class. Have a volunteer
record the unique responses on the board or on a blank transparency on the overhead projector. After all the
groups have shared their findings, have the class agree on a set of findings that best describe the characteristics
of the nature of the roots for the discriminant. Tell the class that for any given quadratic equation there is
either no solution or two solutions. The two solutions could be either a double-root or two different roots.
Now, use a strategy of your choice to model how to use the quadratic formula to solve a quadratic equation.
Have the class follow along with you on the dry–erase boards with the problems you model or have them do a
different problem. Make sure to first write the quadratic equations in general form and identify the three
coefficients a, b, and c. It is very important that students establish a process on how to properly use the
quadratic formula by first writing the quadratic equation in general form and then identifying the coefficients
that will be used in the formula. Students may just want to substitute the coefficients in the formula without
using the correct procedure and then they will not understand how they could arrive at the wrong solutions.
Here are a few examples to model:
• y = x2 + 2x − 3
•
y = x 2 + 3x − 4
•
y = − x 2 + 5x
•
y − x2 − x = 6
•
y − 4x 2 − 1 = 4x
Have the class work in their groups on Exercises 1 through 6. Tell the students that for Exercise 6 they are to
identify the values of a, b, and c from the given quadratic formula and to write the equation for these values.
Bring the class back together and have student volunteers share their results with the rest of the class. You
might want to finish the activity by leading a discussion of the important concepts learned during the activity.
Have a volunteer record the class’ responses on the board or on the overhead transparency. Have the class
record the responses in their student journals.
AIIF
Non–Linear Functions
Lesson 2: The Quadratic Formula
Page 23
Activity 1
SJ Page 8
In this activity, you will be solving quadratic equations using
the quadratic formula to find the values of x when y=0. Make
sure the equations are written in general form before
determining the coefficients a, b, and c.
For the following exercises:
a.
Write the quadratic equation in general form.
b.
Identify the values of a, b, and c.
c.
State the nature of the roots by calculating the
discriminant.
d.
Find all solutions, if any, for the quadratic equation.
1.
x 2 + 3x − 4 = 0 .
a = 1; b = 3; c =– 4.
The discriminant is 25 which is greater than 0; there will be two real distinct roots.
The solutions are x = 1 and x = –4.
2x 2 + 5x + 2 = 0 .
a = 2; b = 5; c = 2.
The discriminant is 9 which is greater than 0; there will be two real distinct roots.
The solutions are x = –2 and x = –1/2.
3x 2 − 8x − 3 = 0 .
a = 3; b = –8; c = –3.
The discriminant is 100 which is greater than 0; there will be two real distinct roots.
The solutions are x = 3 and x = –1/3.
y − x 2 + 4x = 5
a
b.
c.
d.
5.
−b ± b 2 − 4 ac
2a
y = 3 x 2 − 8x − 3
a
b.
c.
d.
4.
Quadratic Formula: x =
y = 2 x 2 + 5x + 2
a
b.
c.
d.
3.
General Form: ax 2 + bx + c = 0
y = x 2 + 3x − 4
a
b.
c.
d.
2.
Standard Form: y = ax 2 + bx + c
− x 2 + 4x − 5 = 0 or x 2 − 4x + 5 = 0 .
a = –1 (or 1); b = 4 (or –4); c = –5 (or 5).
The discriminant is –4 which is less than 0; there are no real roots.
The are no real solutions.
y + 6x − 9 = x 2
a
b.
c.
d.
x 2 − 6x + 9 = 0 .
a = 1; b = –6; c = 9.
The discriminant is 0; there will be one double root.
The double root solution is x = 3.
AIIF
Page 24
SJ Page 9
6.
For the given quadratic formula, identify the values a, b, and c and write the matching quadratic
equation in standard from. Note: use y = ax 2 + bx + c .
a.
x=
−5 ± 52 − 4(1)(3)
2(1)
The values are a = 1, b = 5, and c = 3. The quadratic equation is y = x2 + 5x + 3.
b.
x=
7 ± ( −7)2 − 4(3)( −4)
2(3)
The values are a = 3, b = –7, and c = –4. The quadratic equation is y = 3x2 – 7x – 4
c.
x=
8 ± ( −8)2 − 4( −9)(1)
2( −9)
The values are a = –9, b = –8, and c = 1. The quadratic equation is y = –9x2 – 8x + 1
Non–Linear Functions
Lesson 2: The Quadratic Formula
AIIF
Page 25
Teacher Reference
Activity 2
In this activity, students will write a program for the quadratic formula. Ask the class what was the Setting the
Stage about. Now ask the class if the graphing calculator is somewhat like a computer and could we write a
program for our quadratic formula? Tell the class that there are actually many programs written for the
graphing calculator and yes there are even games for the graphing calculator.
If you haven’t taken the time to show the class how to write a program for the classroom graphing calculator,
you may want to do so now -- not only to show the class the programming keys on the graphing calculator but
also the programming logic as well. This can be considered an activity for the whole class. To create and name
a program, press the following key sequence |Í. The following screen shots show the steps to
create and name a program using the TI-83 or 84 Plus™ graphing calculator.
Enter the name of the program as displayed in the screen shots on the next page, QUADRTIC, or some other
name you prefer. There are three menus that are used for writing a program: CTL (program control and logic),
I/O (input/output), and EXEC (execute an existing program; we'll not use this one). The relational and logic
operators are under the TEST button (y). The CTL menu consists of the following (only the commands
used most often are listed below):
1:If – Creates a conditional test.
2:Then – Executes commands when If is true.
3:Else – Executes commands when If is false.
4:For( – Creates an incrementing loop.
5:While – Creates a conditional loop.
6:Repeat – Creates a conditional loop.
7:End – Signifies the end of a block. Used with If–Then, Else, For, While, and Repeat
8:Pause – Pauses program execution.
9:Lbl – Defines a label.
0:Goto – Goes to a label.
E:Return – Returns from a subroutine. Within the main program, Return stops execution and returns to the
home screen.
F:Stop – Stops execution.
If you have CtlgHelp application, run it to have catalogue help available. Catalogue help will give you the
format of any command or operator by pressing the à key.
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Page 26
The I/O menu consists of the following (only the commands used most often are listed below):
1:Input – Enters a value.
2:Prompt – Prompts for entry of variable values.
3:Disp – Displays text, value, or the home screen.
4:DispGraph – Displays the current graph.
5:DispTable – Displays the current table.
6:Output – Displays text at a specified position.
8:ClrHome – Clears the display.
Make sure the class takes into consideration the discriminant when writing the program. The purpose of this
activity is to show the class the power of the graphing calculator. Adjust the program as necessary for the
classroom graphing calculator. You may want to skip the first two screen shots and start with the third, which
displays a message about entering the values for a, b, and c. Note: View the screen shots below carefully. Some
of the screen shots have the last line repeated in the next shot. This is to show connectivity because some of the
commands can't be totally displayed and to help determine what was the last command entered.
Have the students continue working in their groups to write the program. Suggest naming the program
QUADRTIC or something similar. It is probably best that the class test their programs on the exercises from
Activity 1 or 2 or some quadratic equations you may have in a textbook. Bring the class together and have
volunteers share their program with the class, using the graphing calculator view screen. Determine if the class
enjoyed this activity. Tell the students that this experience can be useful in other courses such as physics or
other mathematics courses.
Non–Linear Functions
Lesson 2: The Quadratic Formula
AIIF
Page 27
Activity 2
SJ Page 10
In this activity, your teacher will guide you through writing a program for the quadratic formula on the
classroom graphing calculator. Test your program on the first activity. Things you will need to pay attention to
in your program are:
•
•
•
•
•
The discriminant
Programming logic
Data input
Data output
Calculations using the quadratic formula
Use the supplemental exercises below to further test your program by solving for x when y=0. Round your
answers to 3 decimal places.
1.
y = 3x 2 − 6 x − 9
The solutions are 3 and –1.
2.
y = 12 x 2 + x − 5
The solutions are approximately 0.605 and –0.689
3.
y = 13x 2 + 6 x + 1
There are no real solutions.
4.
y = 0.25x 2 + 6 x + 4
The solutions are approximately –0.686 and –23.314.
5.
y = x 2 − 12 x + 36
The double root solution is 6.
6.
Can you think of any improvements in the program you wrote?
Answers may vary.
AIIF
Page 28
Teacher Reference
Activity 3
In this activity, students will continue using the quadratic formula to solve application problems. Discuss what
the class might need to consider to solve quadratic equations that wasn’t necessary in the first activity. The key
idea here is that students will determine if any solution does not make sense in the real–world application. For
instance, if an answer has a negative value but that negative value does not make sense in the application
problem, then the students will need to discard that particular answer.
For example, in dropping a ball from a building it is determined that the ball hit the ground at times t = 6 and
t = –3, then the –3 value must be discarded since time is not measured using negative numbers. Ask the class,
"What is the height of the ball when it hits the ground?" Students should know that the height is zero, so to
solve an equation involving something hitting the ground is the same as solving an equation in general form.
Also, discuss with the class the concept of break–even point. The break–even point is where the revenue
equals the costs. No money is made (positive profit) and no money is lost (negative profit). Other issues to
discuss include:
• Farthest distance traveled involves the time it takes for something to hit the ground once it is thrown or
shot skyward.
• Gravity plays an important roll in projectile motion (baseball thrown, model rocket launched, etc).
• If we have to find how long it takes until an object is a certain height, then we set the function
representing the position to the given height. For example, if s(t) = –16t2 + 25t + 10 and we need to find
the time that the object has a position, or height, of 50 feet, then the equation would be 50 = –16t2 + 25t
+ 10. The students would then have to rewrite the equation in general form to find t.
Have the students continue working in their groups, but begin working with a partner, for Exercises 1 through
4. After each pair has completed the exercises, have the pair compare answers with the other pair in their
group and settle any discrepancies in their answers either amongst themselves or with other groups. Have
volunteers share their results with the class.
AIIF
Non–Linear Functions
Lesson 2: The Quadratic Formula
Page 29
Activity 3
SJ Page 11
In this activity, you will continue to use the quadratic formula to
solve quadratic equations for real-world applications. Use the same
process from Activity 1 to find your solutions (write the equation in
general form; identify the coefficients a, b, and c.) Make sure your
answers make sense for the real-world application problem.
Break Even Point – The point
where the revenue, R(x), equals
the cost, C(x). Symbolically,
R(x) = C(x).
1.
You have a part–time job working for a local machine shop. The owner plans to make a certain product
to sell. The product's costs are related by the function C ( x ) = 6250 + 50 x + x 2 and the owner knows he can
sell the product for $325.00 each, giving him a total revenue of R( x ) = 325x , where x represents the
number of items produced. The owner would like you to find the break-even points so he can
determine the number of the product items he should produce each week.
The break-even points are (25, 0) and (250, 0).
2.
A ball is thrown downward from the top of a building into a river. The height of the ball from the river
can be modeled by H (t ) = −16t 2 − 15t + 600 , where t is the time, in seconds, after the ball was thrown.
How long after the ball is thrown is it 75 feet above the river? How long, to the nearest tenth of a
second, does it take the ball to land in the river?
It takes the ball 5.3 seconds until it is 75 feet above the river. It takes the ball about 5.7 seconds to land
in the river.
3.
It takes a 2004 Corvette 4.3 seconds to accelerate from 0 to 60 miles per hour. The same car can do the
quarter mile, 1320 feet, in 12.7 seconds. The displacement function can be described by the equation
s(t ) = 4.09t 2 + 51.99t .
4.
a.
How far has the Corvette traveled after 4.3 seconds, to the nearest foot?
The Corvette has traveled 299 feet in 4.3 seconds.
b.
How long does it take the Corvette to travel half a mile? Round your answer to the nearest tenth
of a second. Note: A mile is 5,280 feet.
It takes the Corvette approximately 19.8 seconds to travel half a mile.
The Coast Guard is testing two rescue flares from two competing companies. The Coast Guard plans to
sign a contract with the company whose rescue flare travels the farthest. The Coast Guard fires the two
flares into the air over the ocean. The paths of the flares are given by:
Company A: y = −0.000253x 2 + x + 15
Company B: y =
−x 2 56
+ x + 15
243 3
where y is the height and x is the horizontal distance traveled. Determine which flare the Coast Guard
should purchase by substituting y = 0 into each equation and finding x. What does the constant 15
represent in each equation?
Company A’s flare travels horizontally about 3968 feet, while company B’s flare travels horizontally
about 4537 feet. The Coast Guard should purchase the flare from company B. The constant 15
represents the height from which the flare was fired.
AIIF
Page 30
SJ Page 12
Practice Exercises
For Exercises 1 through 3:
a.
b.
c.
d.
Write the quadratic equation in general form.
Identify the values of a, b, and c.
State the nature of the roots by calculating the discriminant.
Find all solutions, if any, for x when y=0 for the quadratic equation.
Round all answers to the nearest tenth.
1.
y = 11x 2 − 10 x − 1
a.
b.
c.
d.
2.
y = −3x 2 + 5x + 12
a.
b.
c.
d.
3.
−3x 2 + 5x + 12 = 0
a = –3; b = 5; c = 12.
The discriminant is 169 which is greater than 0; there will be two real distinct roots.
The solutions are x = 3 and x = –4/3.
y = −2 x 2 − 8x − 8
a.
b.
c.
d.
4.
11x 2 − 10x − 1 = 0
a = 11; b = –10; c = –1.
The discriminant is 144 which is greater than 0; there will be two real distinct roots.
The solutions are x = 1 and x = –1/11.
−2x 2 − 8x − 8 = 0
a = –2; b = –8; c = –8.
The discriminant is 0; there will be a double root.
The solutions are x = –2 and x = –2.
Cox’s formula for measuring velocity of water draining
from a reservoir through a horizontal pipe is
1200 HD
= 4 v 2 + 5v − 2 , where v represents the velocity
L
H
of the water in feet per second, D represents the
diameter of the pipe in inches, H represents the height
of the reservoir in feet, and L represents the length of
pipe in feet. How fast is water flowing through a 30
foot long pipe with diameter of 24 inches that is
draining from a pond with a depth of 30 feet? Round
your answer to the nearest tenth of a foot per second.
The velocity of the water in the pipe is approximately 84.2 feet per second.
D
L
AIIF
Non–Linear Functions
Lesson 2: The Quadratic Formula
Page 31
5.
SJ Page 13
A ball is thrown upward with an initial velocity of 146 feet per second from a height of 7 feet. How long
does it take the ball to hit the ground? The equation for projectile motion is s(t) = –16t2 +v0t + h0, where s
is the height of the projectile in feet, t is the time in seconds, v0 is the initial velocity, and h0 is the initial
height. Round your answer to the nearest tenth of a second.
It takes the ball 9.2 seconds to hit the ground.
6.
For the given quadratic formula, identify the values a, b, and c and write the quadratic equation from
these values.
a.
x=
−11 ± (11)2 − 4(5)(6)
2(5)
The values are a = 5, b = 11, and c = 6. The quadratic equation is y = 5x2 + 11x + 6.
b.
x=
12 ± ( −12)2 − 4( −2)( −19)
2( −2)
The values are a = –2, b = –12, and c = –19. The quadratic equation is y = –2x2 – 12x – 19.
For part b. above, will the quadratic equation have any real solutions? Explain.
c.
For part b. above there will not be any real solutions because the discriminant has a value of
–8.
7.
Find the mistake below and correct it.
x 2 − 13x = 7
2
x − 13x = 7
2
=
−( −13) ± ( −13) − 4(1)(7)
2(1)
13 ± 169 − 28
2
13 ± 141
=
2
13 ± 11.9
≈
2
13 + 11.9
≈
= 12.45
2
and
13 − 11.9
≈
= 0.55
2
=
The equation was not written in general form first.
x 2 − 13x − 7 = 0
=
−( −13) ± ( −13) 2 − 4(1)( −7 )
2(1)
13 ± 169 + 28
2
13 ± 197
=
2
13 ± 14
≈
2
13 + 14
≈
= 13.5
2
and
13 − 14
≈
= −0.5
2
=
AIIF
Page 32
SJ Page 14
Outcome Sentences
I know that the discriminant portion of the quadratic formula is used to
I know that the quadratic equation must be in ___________________________________________form to be
When solving real-world applications using the quadratic formula
The part of the quadratic formula I don’t understand is_______________________________________because
Non–Linear Functions
Lesson 2: The Quadratic Formula
AIIF
Page 33
Teacher Reference
Lesson 2 Quiz Answers
1.
2.
a.
x 2 + 15x + 25 = 0
b.
a = 1; b = 15; c = 25.
c.
The discriminant is 125 which is greater than 0; there will be two real distinct roots.
d.
The solutions are x ≈ –1.91 and x ≈ –13.09.
a.
− x 2 − 7 x − 13 = 0 or x 2 + 7 x + 13 = 0
b.
a = –1 or 1; b = –7 or 7; c = –13 or 13.
c.
The discriminant is –3 which is less than 0; there will be no real roots
d.
There are no real solutions
AIIF
Page 34
Lesson 2 Quiz
Name:
For problems 1 and 2:
a.
Write the quadratic equation in general form.
b.
Identify the values of a, b, and c.
c.
State the nature of the roots by calculating the discriminant.
d.
Find all solutions, if any, for x when y=0 for the quadratic equation.
1.
y = x 2 + 15x + 25
2.
y = −x 2 − 7 x − 13
Non–Linear Functions
Lesson 3: Graphing Quadratic Functions and Their Applications
AIIF
Page 35
Lesson 3: Graphing Quadratic Functions and Their Applications
Objectives
• Students will understand that functions are used to model and analyze real-world applications and
quantitative relationships.
• Students will understand that functions come in many different forms and are often needed to solve or
simplify abstract ideas.
• Students will see how to use technology to understand, simplify, and solve complicated abstract ideas.
Essential Questions
• How can quadratic functions and applications of quadratic functions be graphed?
Tools
• Student Journal
• Setting the Stage transparencies
• Dry–erase boards, markers, erasers
• Graphing calculators
Warm Up
• Problems of the Day
Number of Days
• 3 days
Vocabulary
Symmetry
Vertical line symmetry
y-coordinate
Seven-pin polygon
Vertical translation
Vertex
Quadratic regression
Horizontal translation
x-coordinate
Congruent
AIIF
Page 36
Teacher Reference
Setting the Stage
Before placing the Setting the Stage transparency 1 on the overhead projector, lead a discussion with the class
about symmetry. Ask the class, “What does it mean if an object has symmetry?” Have a volunteer list the class
responses on the board or on the overhead projector. After students have given several responses, ask, “What
types of symmetry are there?” Have the same volunteer list the class responses, but have the responding
students come to the board to demonstrate the type of symmetry they suggested. Make sure the class agrees
with the demonstration before moving on to another type.
Now place the Setting the Stage transparency 1 on the overhead projector. Tell the class that the type of
symmetry they will be doing is vertical line symmetry. Remind the class that line symmetry means that an
object can be folded, or reflected, so that the two parts are congruent. Vertical line symmetry means folding or
reflecting the object about a vertical line. Have the class work in groups of four. Ask the class which letters of
the alphabet have vertical line symmetry. You might want to use the letter A as an example by drawing a
vertical line down the middle of the letter. Tell the students they can either visualize the symmetry or fold the
letter in half to determine vertical line symmetry. Give the groups a couple of minutes to determine which
letters have vertical line symmetry. Have each group give a letter that has vertical line symmetry while a
volunteer circles the given letters on the overhead transparency. Ask the class if all letters that have vertical
line symmetry have been circled. If they haven’t, ask groups to name the remaining letters.
Next, place the Setting the Stage transparency 2 on the overhead projector. Tell the class that the seven dots are
arranged in a hexagonal pattern. Tell the class that a seven–pin polygon is a closed shape made by joining the
pins, or dots, with straight lines. Draw the following polygons on the board. Tell the students that their
objective is to draw as many seven–pin polygons with vertical line symmetry as they can. Give the class about
two or three minutes to complete the activity. Have someone from each group draw a seven–pin polygon with
vertical line symmetry on the transparency.
Here are some examples of seven–pin polygons that have vertical line symmetry:
Non–Linear Functions
Lesson 3: Graphing Quadratic Functions and Their Applications
AIIF
Page 37
Setting the Stage
Transparency 1
ABCDE
FGHIJK
LMNOP
QRSTU
VXYZ
AIIF
Page 38
Setting the Stage
Transparency 2
Seven Pin Polygons
Non–Linear Functions
Lesson 3: Graphing Quadratic Functions and Their Applications
AIIF
Page 39
Teacher Reference
Activity 1
Place Activity 1 transparency 1 on the overhead projector. Have the class draw the plotted points and
corresponding curved graph on their dry–erase boards as best they can. Tell the class that they can
approximate the points. Have a volunteer model vertical line symmetry while the class uses dry–erase boards
and assists the volunteer as needed. Tell the students to use their knowledge of vertical line symmetry to
reflect their plotted points across the y–axis and then draw the corresponding curved graph. Lead a discussion
with the class about the vertical line symmetry they did for the Setting the Stage and how it can be applied to a
set of points and corresponding graph. Have the students hold up their dry–erase boards and visually inspect
their results. Ask, “Does the graph have a minimum or maximum value? Explain why. Also, what is the
minimum or maximum value?” The class should determine the answer by how the graph opens. Remember a
graph that opens upward has a minimum value and a graph that opens downward has a maximum value.
Place Activity 1 transparency 2 on the overhead projector with the bottom table covered. Tell the class to plot
the points from the table on their dry–erase boards while a volunteer plots the points on the board. Lead a
class discussion about the vertical line of symmetry at x = 4 and how it might affect them in determining the
reflected points. You also might want to ask the class which point was not reflected about the vertical line and
why. Have the class draw the graph of the plotted points. Ask the class if they recognize the graph. The class
should notice the shape of the graph from the quadratic functions they graphed in Lesson 1. You can either
discuss or point out to the class that x–values that are the same distance from the minimum value have the
same y–value. This is a very important concept for vertical line symmetry for quadratic functions. Ask the class
“Without plotting the points and drawing the graph, is there a way to determine if we have a minimum or
maximum value? Explain why. Also, what is the minimum or maximum value?” Students should base their
answers on the y–values. If the y–values are decreasing and then increasing, the graph opens upward.
Likewise, if the y–values are increasing then decreasing, the graph opens downward.
Uncover the bottom table on Activity 1 transparency 2. Continue the discussion about the point in the table
that won’t be reflected about an axis. Have a third volunteer plot the table values on the board and reflected
points while the class plots and reflects on their dry–erase boards. Ask the class what is different about this
graph compared to the others they have done.
Place Activity 1 transparency 3 on the overhead projector. Have volunteers state the equation of the vertical
line of symmetry for each graph and table. Have the class agree on these equations. Note: These are four
separate problems. Ask the students how they determined the equation for the vertical line of symmetry. Also,
ask the students, “Do the graphs or data tables have a minimum or maximum value? Explain why. What is the
minimum or maximum value of each graph and data table?” and "How is the minimum or maximum value
related to the vertical line of symmetry?"
AIIF
Page 40
Some of the important concepts that the students should understand, either in this pre-activity modeling and
discussion or by the time they have completed the exercises in Activity 1, are:
•
•
•
•
•
Opposite x-values have the same y-value (for vertical line of symmetry being the y–axis.)
Opposite x-values are the same distance from the minimum or maximum x-value (for vertical line of
symmetry not being the y–axis.)
The left side of the graph is a mirror image of the right side and vice–versa.
The x–coordinate of the minimum or maximum point determines the equation for the vertical line of
symmetry.
A graph that opens upward has a minimum value and a graph that opens downward has a maximum
value.
Have the class work in pairs on Exercises 1 through 4. Have each pair get together with another pair after
completing the exercises. Within these groups, have student pairs share their results. If there are any
discrepancies in their results, have the group check with other groups about their results and agree on the final
results for each exercise. Then have the groups share their findings.
AIIF
Non–Linear Functions
Lesson 3: Graphing Quadratic Functions and Their Applications
Page 41
Activity 1
Transparency 1
y
3
2
1
x
–6
–3
3
–1
–2
–3
6
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Page 42
Activity 1
Transparency 2
Vertical Line of Symmetry: x = 4
x
–6
–4
–2
0
2
4
y
100
64
36
16
4
0
Vertical Line of Symmetry: x = –2
x
–6
–5
–4
–3
–2
y
–14
–7
–2
1
2
AIIF
Non–Linear Functions
Lesson 3: Graphing Quadratic Functions and Their Applications
Page 43
Activity 1
Transparency 3
x
2
3
4
5
6
7
8
y
5.25
4
3.25
3
3.25
4
5.25
x
–13
–10
–7
–4
–1
2
5
y
–496
–226
–64
–10
–64
–226
–496
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Page 44
Activity 1
SJ Page 15
In this activity, you will use your knowledge of the vertical line of symmetry to plot points, draw a graph, and
find the equation for the vertical line of symmetry.
1.
Using the given dashed vertical line of symmetry, plot and draw the missing half of the graph. Write the
equation for the vertical line of symmetry. State whether the graphs have a minimum or a maximum
value and explain why.
a.
2.
b.
The equation for the vertical line of
The equation for the vertical line of
symmetry is x = –2. The graph opens
x = 3. The graph opens downward so it has
upward so it has a minimum.
a maximum.
Complete the tables below using the values in the table along with the equation for the vertical line of
symmetry. Plot the points in the table, draw the graph, and draw the vertical line of symmetry. State
the equation of the vertical line of symmetry, whether the data tables have a minimum or a maximum
value, and explain why.
a.
Vertical line of symmetry: x = 3.
It has a maximum value.
x
–6
–3
0
3
6
9
12
b.
y
–73
–28
–1
8
–1
–28
–73
Vertical line of symmetry: x = –1/2.
It has a minimum value.
x
–5
–7/2
–2
–1/2
1
5/2
4
y
71/2
13
–1/2
–5
–1/2
13
71/2
AIIF
Non–Linear Functions
Lesson 3: Graphing Quadratic Functions and Their Applications
Page 45
SJ Page 16
3.
Write the equation for the vertical line of symmetry for the given graphs. State whether the graphs
have a minimum or a maximum value and explain why.
a.
b.
The equation for the vertical line of
symmetry is x = 0. The graph opens
up and has a minimum.
4.
The equation for the vertical line of
symmetry is x = 9/2 or 4.5. The graph
opens down so it has a maximum.
Write the equation for the vertical line of symmetry for the data tables below. State whether the data
tables have a minimum or a maximum value and explain why. Also state the minimum or maximum
value.
a.
b.
x
–18
–14
–10
–6
–2
2
6
y
10
0
–6
–8
–6
0
10
The equation for the vertical line of
symmetry is x = –6. The y-values in
data table are decreasing and then
increasing so there is minimum value
of –8.
x
0
2
4
6
8
10
12
y
–7
–2
1
2
1
–2
–7
The equation for the vertical line of
symmetry is x = 6. The y-values in the data
the table are increasing and then decreasing
so there is a maximum value of 2.
AIIF
Page 46
Teacher Reference
Activity 2
In this activity, students will continue to explore the line of symmetry and will be introduced to the term
vertex. Using a strategy of your choice, model how to find the equation of the line of symmetry
for y = x 2 − 4 x − 2 . Use the graphing calculator, view screen, and the techniques from the Lesson 1 to find the
minimum value for the quadratic equation. Have the students write the equation for the line of symmetry on
their dry–erase boards and hold up the boards. Ask the class members what they wrote for the equation for the
line of symmetry. Make sure the class agrees on this equation. Show the class how to use the line of symmetry
to find values of x that have the same y–value. Tell the class that the distance between the two x–values that
have the same y–value is the same on each side of the vertical line of symmetry. This technique makes it easy
to use symmetry to graph quadratic functions because we need to find only the minimum or maximum value
and a few points on one side of the minimum or maximum point and their opposites by symmetry.
For example, from the quadratic equation above, we found the minimum to be the point (1, –6). If we can find
the y–values for x equal to 0, –2, and –4, we could then use symmetry to get the same y–values for the x–values
of 2, 4, and 6 (these three values of x are the same distance from the vertex as the first three x values). Tell the
class that this minimum or maximum value has a special name, it is called the vertex. Ask if anyone has
encountered this word before. They might have heard it as part of an angle or a regular polygon.
Lead the following dialogue with the class:
“We saw in Lesson 1 that quadratic functions of the form y = ax 2 , where a is any positive number, that the
graph opened upward and the vertex point (a minimum) was the origin (0, 0). If a was a negative number
for y = ax 2 , the graph opened downward and the vertex point (a maximum) was the origin (0, 0). We also saw
that if the quadratic function was of the form y = ax 2 + k , where k is any positive number then the vertex point
(minimum or maximum) was at (0, k). If the quadratic function was of the form y = ax 2 − k , where k is any
positive number then the vertex point (minimum or maximum) was at (0, –k). Likewise, quadratic functions of
the form y = a( x − h )2 , where h is any positive number, have a minimum or maximum value of (h, 0). And, if the
quadratic functions were of the form y = a( x + h )2 , where h is any positive number, then the vertex point
(minimum or maximum) was at (–h, 0). We know that h caused a horizontal translation of the vertex point
while k caused a vertical translation. Using h and k together we can translate the vertex both horizontally and
vertically at the same time.
Now, what if we didn’t have a graphing calculator and we had to find the minimum or maximum value
to y = x 2 − 4 x − 2 . How could we find the minimum or maximum value with out creating a table of values or a
rough sketch of the graph? Are there any special formulas that could help us? It turns out there is such a
−b
formula. The x–coordinate of the vertex can be found by using: x =
and the y–coordinate can be found by
2a
substituting the x–value into the equation, or namely y = f ( x ) , where f(x) = ax 2 + bx + c . Tell the class that the
⎛ −b ⎛ −b ⎞ ⎞
coordinates for the vertex are ⎜ , f ⎜ ⎟ ⎟ , but the equation must be written in the standard form
⎝ 2a ⎝ 2a ⎠ ⎠
2
y = ax + bx + c . Note that the number in front of the x squared term is a, the number in front of the x term is b,
and the constant term is c. Does c help us find any particular point on the graph of a quadratic equation?” The
class should realize that a constant for any equation, linear or non–linear, generally represents the y–intercept.
Non–Linear Functions
Lesson 3: Graphing Quadratic Functions and Their Applications
AIIF
Page 47
Model, or have a volunteer model, finding the vertex for y = x 2 − 4 x − 2 while the class parallels
with y = x 2 + 6 x + 8 using dry–erase boards, but not the graphing calculator. Note that a = 1, b = –4 and a = 1, b =
6, respectively. Using the formula for the vertex, the volunteer and the class should have calculated the vertex
to be (2, –6) and (–3,–1), respectively. Have another volunteer model finding the vertex for the quadratic
function y = 4 x 2 − 8x + 2 (x = 1 and y = –2) on the board or overhead projector while the class finds the vertex
for y = −2 x 2 − 12 x + 3 (x = –3 and y = 21). Model as many quadratic functions as needed by using the following
examples.
Examples
y = x 2 + x + 5 , vertex is (–1/2, 19/4) or (–0.5, 4.75).
y = 4 x 2 − 24 x + 8 , vertex is (3, –28).
1 2
3
x − 2 x + , vertex is (2, –1/2) or (2, –0.5).
2
2
y = −5x 2 − 15x − 7 , vertex is (–3/2, 17/4) or (–1.5, 4.25).
y=
Have the class work in pairs on Exercises 1 through 5. Have volunteers share their results and the class agree
on the results. If any students has a problem with any exercise, have another volunteer model his/her results
on the board or overhead projector.
Now ask the class, “What are the techniques you can use to graph any quadratic function without a graphing
calculator or other technological tool?” Have a volunteer record the list of techniques, given by the class, either
on the board or overhead projector. Each student should record this list in their student journals. Their list
should include the following:
•
•
•
•
Vertex
y–intercept
x–intercept(s) (discriminant)
Line of symmetry
The class should now be able to graph any quadratic function along with applications of quadratic functions.
Using either the above list or the list the students compiled, model graphing the quadratic functions above
while the students use their dry–erase boards to graph the same equations.
Have pairs of students get together to form groups of four for Exercises 6 and 7. Tell the class to use its
understanding of quadratic functions to find the functions for the given graphs. You may want to model how
to do this with Exercise 6. The students should be able to use their knowledge of the vertical line of symmetry,
vertex, and the various forms of a quadratic function to do this. Bring the class together and have groups share
their results on the board.
AIIF
Page 48
SJ Page 17
Activity 2
In this activity, you will be determining specific characteristics
of quadratic functions and real–world problems involving
quadratic functions and then graphing the quadratic functions
from the characteristics. In the following exercises you will
need to:
a.
b.
c.
d.
e.
1.
d.
e.
2.
a = –1; b = 4; c = –5
Coordinates of the vertex are (2, –1)
The y-intercept is (0, –5); there are no xintercepts
Line of symmetry is x = 2
Several points on each side of the vertex may
vary. Sample points are (–1, –10), (5, –10), (–4, –
37), and (8, –37)
y = 3 x 2 − 8x − 3
a
b.
c.
d.
e.
3.
Identify the values of a, b, and c
Vertex coordinates
All intercepts
Line of symmetry
Several points on either side of the vertex
y = −x 2 + 4x − 5
a
b.
c.
a = 3; b = –8; c = –3
Coordinates of the vertex are (4/3, –25/3)
The y-intercept is (0, –3); x-intercepts are (3, 0)
and (–1/3, 0)
Line of symmetry is x =4/3
Several points on each side of the vertex may
vary. Sample points are (2, –7), (2/3, –7), (–2, 25),
and (14/3, 25)
y = 2 x 2 − 5x + 2
a
b.
c.
d.
e.
−b
2a
⎛ −b ⎞
Vertex – y coordinate: y = f ⎜ ⎟
⎝ 2a ⎠
−b
Line of Symmetry: x =
2a
Vertex – x coordinate: x =
a = 2; b = –5; c = 2
Coordinates of the vertex are (1.25, –1.125)
The y-intercept is (0, 2); x-intercepts are (0.5, 0)
and (2, 0)
Line of symmetry is x =5/4 or x = 1.25
Several points on each side of the vertex may
vary. Sample points are (0, 2), (–1, 9), (2.5, 2), and
(3.5, 9)
Quadratic Formula: x =
Discriminant: b 2 − 4 ac
−b ± b 2 − 4 ac
2a
AIIF
Non–Linear Functions
Lesson 3: Graphing Quadratic Functions and Their Applications
Page 49
4.
SJ Page 18
Photosynthesis is the process in which plants use the energy from the sun's rays to convert carbon
dioxide to oxygen. The intensity of light is measured in lumens. Let R be the rate that a certain plant
uses to convert the sun's light energy. Let x be the intensity of the light. The plant converts the carbon
dioxide at a rate according to the equation R = 240 x − 80 x 2 . Sketch the graph of this equation and
determine the intensity that gives the maximum rate of photosynthesis. State the domain which makes
sense for the application.
The intensity of light that gives maximum rate of
photosynthesis is 1.5 lumens with a maximum rate of
180. The line of symmetry is x = 3/2 (or x = 1.5); yintercept is (0, 0); x-intercepts are (0, 0) and (3, 0). Some
points on either side of the vertex are (1, 160), (2, 160),
(0.5, 100), and (2.5, 100). Note: y-axis has a scale of 10.
5.
The cost function to make a certain product
is C ( x ) = 0.2 x 2 − 10 x + 360 . The revenue function for
the same product is given by R( x ) = −0.2 x 2 + 50 x .
a.
b.
Graph the cost and revenue cost functions on
the same set of axes.
What level of production will produce the
maximum revenue? What is the maximum
revenue?
The level of production that maximizes
revenue is 125 units. The maximum revenue is
$3125
c.
What level of production will produce the
minimum cost? What is the minimum cost?
The level of production that minimizes cost
is 25 units. The minimum cost is $235
d.
Graph the profit function (profit = revenue
minus cost) on a separate set of axes.
e.
What level of production will produce the
maximum profit? What is the maximum
profit?
The level of production that maximizes
profit is 75 units. The maximum profit is
$1890
4000
2000
30
60
90
120
150
30
60
90
120
150
1600
800
AIIF
Page 50
SJ Page 19
6.
The graph below represents the profit function for a company that produces widgets. Find the
equation of the profit function P(x). Note: You should be able to determine the value of c from the
graph. Also, use the coordinates of the vertex to find a and b. Use –b/2a for the x-coordinate and solve
for b in terms of a and substitute this value into y = ax2 + bx + c to find a and then b.
The value of a = -1, b = 100, and c = –
1200. From the graph we see that the
y–intercept is –1200. Hence c = –1200.
To find "a" and "b", use vertex
formula to get x = –b/2a or b =–2ax.
Looking at the graph we see that the
x-value of the vertex is 50 and the
y-value is 1300. Using the
equation 1300 = a(50) 2 − 2a(5) − 1200 ,
we get that a = –1 and then b = 100.
The quadratic equation of the graph
is P( x ) = − x 2 + 100x − 1200
7.
200
–200
10
A town is having a parade and celebration for its high school marching band. The school’s marching
band recently marched in Macy’s Annual Thanksgiving Day Parade®. This was the first time the
marching band is being honored for its hard work and achievement in the state competition. The town
wants to hang a banner on a steel cable between its two tallest buildings -- each 100 feet tall. The
distance between the two buildings is 50 feet. The weight of the banner caused the bottom of the
banner to be 20 feet lower than the top of the building. Assume the bottom of the banner is parabolic in
shape. What is the quadratic function that represents the lower portion of the banner?
Answers may vary. A sample response might be: “The quadratic equation that represents the bottom
portion of the banner is y = 0.032x 2 + 80 .” Draw a coordinate system such that the position of the
origin is on the ground halfway between the buildings.
Non–Linear Functions
Lesson 3: Graphing Quadratic Functions and Their Applications
AIIF
Page 51
Teacher Reference
Activity 3
In this activity, students complete quadratic data modeling by fitting data to quadratic equations. Remind the
students that in the Linear Functions unit, they did linear data modeling. They drew the line of best fit and then
determined the equation for the line they drew. Also, the last activity had the students determining quadratic
functions from graphs of parabolas.
Place the Activity 3 transparency on the overhead. Ask the class which scatter plots most resemble a parabola
or have a quadratic trend. Students might not think that graph D on the right is a parabola, but if they look
close and draw the parabola of best fit they could possibly see a portion of the arc of the parabola. Have the
class answer Exercise 1 based on which scatter plots they felt had a quadratic trend. Have volunteers draw the
best fit parabola on the graphs that most resemble a parabola; the rest of the class will work in pairs from the
last activity to draw the best fit parabola so they can determine the quadratic function. Walk around the room
and assist the class on Exercises 2 and 3 as needed and give blank grid transparencies to pairs of your choice to
have them share with the rest of the class.
If necessary, review with the class how to enter data into the graphing calculator. The key sequence, for
quadratic regression for the TI–83 or 84 Plus™ family of calculators, is … ~ ·. QuadReg is displayed to
the home screen. By default, QuadReg uses L1 and L2 as the lists for x and y, respectively. If the data has been
entered in different lists then press the necessary keys to get the appropriate lists. For example, if the lists for x
and y were L3 and L4, respectively, then press y  (L3) y ¶ (L4). Pressing the Í key will execute the
QuadReg command and display the results to the home screen. Don't worry about any diagnostics for the
quadratic function; we won't be discussing diagnostics with quadratics as it is beyond the scope of this lesson.
QuadReg requires at least three sets of ordered pairs. The following screen shots show selecting QuadReg,
having it displayed to the home screen, and the results of executing the command.
You also may opt to show the class, if you haven't before, how to do a scatter plot with their data sets. After
the data have been entered using the … key, press the y o keys (,) to display the STAT PLOTS
menu. Select the desired stat plot (use the first one by default). Pressing the Í key to turn on the stat plot,
select the desired Type: (first type is suggested), XList and YList default to L1 and L2 respectively. Select the
Mark: the square mark is suggested. The screen shots below show the steps on doing a scatter (stat) plot.
AIIF
Page 52
To have the best fit parabola sent to Y1 in the Y= editor to display the scatter plot and the best fit parabola at
the same time, follow this key sequence: use the steps above to display QuadReg to the home screen; press
 ~ (to select Y–VARS) Í (or À to select 1:Function…) À (or Í to select 1:Y1). Your home screen
should have QuadReg Y1 displayed. Pressing the Í key will execute the command and the results will be
displayed to the home screen and the equation to Y1 (see screen shots below).
Have the class continue to work in their pairs on Exercises 4 through 6. Student pairs can check their results
with one or more other pairs. Have volunteers share their results using the calculator view screen.
AIIF
Non–Linear Functions
Lesson 3: Graphing Quadratic Functions and Their Applications
Page 53
Activity 3 Transparency
A
B
C
D
AIIF
Page 54
SJ Page 20
Activity 3
In this activity, you will use your knowledge and understanding of quadratic functions to do quadratic
regression on scatter plots and data sets. In the last activity, you wrote quadratic functions from graphs. In the
Linear Functions unit, you drew the best fit line for a scatter plot and determined the equation for the line of
best fit. In this activity, you will use the concepts and skills developed in the Linear Functions unit to draw the
best fit parabola for given graphs and then determine the equation for the parabola you drew.
1.
Which scatter plots below seem to have a quadratic trend? Scatter plots A, C, and D seems to have a
quadratic trend.
A
B
C
D
2.
Draw a best fit parabola for the scatter plots you determined had a quadratic trend in Exercise 1.
Answers will vary
3.
Determine the quadratic functions from the best fit parabolas you drew in Exercise 2.
Answers will vary
AIIF
Non–Linear Functions
Lesson 3: Graphing Quadratic Functions and Their Applications
Page 55
SJ Page 21
In the Linear Functions unit, you learned to use the graphing calculator to determine the equation for the best fit
line from sets of data. We called this linear regression. The graphing calculator can also be used to determine the
equation for the best fit parabola from sets of data. We call this quadratic regression. Follow your teacher's
instructions on how to use the graphing calculator to determine the quadratic function from sets of data.
4.
The table below shows the U. S. population distributed by age (x) and percentage (y).
x
y
5.
Under 5
1
7.4%
5 to 17
2
18.2%
18 to 44
3
43.2%
45 to 64
4
18.6%
65 and over
5
12.6%
a.
Determine the equation for the parabola of best fit. Round the values of a, b, and c to three
decimal places.
The equation for the best fit parabola is y = –5.943x2 + 36.737x – 24.84
b.
Use your graphing calculator to create a
scatter plot and graph of the data and
sketch the scatter plot and graph on the
set of axes.
The graphing calculator plot and graph
is:
c.
How well does the graph of the best fit parabola fit the data?
Sample response: The graph of the best fit parabola does not reach the vertex, maximum value,
of the data.
The students of Mr. G's class were told to record the number of hours spent studying for their
mathematics test. For each student, Mr. G wrote an ordered pair (x, y). The x-value represented the
number of hours the student spent studying and the y-value represented the student’s test score.
(0.5, 40), (9.3, 75), (8.4, 80), (0.5, 56), (1.0, 60), (8.2, 83), (7.6, 87), (1.0, 47), (1.4, 48), (7.0, 91), (6.5, 94), (1.5,
63), (2.0, 73), (6.2 98), (5.5, 100), (2.3, 78),(2.4, 83), (5.4, 97), (5.4, 98), (2.5, 77), (2.6, 83), (5.2, 95), (5.1, 85),
(3.0, 88), (3.0, 86), (4.9, 94), (4.2, 93), (3.5, 91), (3.5, 90), (3.7,89).
a.
Use your graphing calculator to create a scatter plot. Does the data seem to model a quadratic
equation? Explain. Sample response: Yes, the data seems to model a quadratic equation
because the scatter plot is shaped like a parabola.
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Page 56
SJ Page 22
6.
b.
Determine the equation for the parabola of best fit. Round the values of a, b, and c to three
decimal places.
The equation for the best fit parabola is y = –1.944x2 + 21.819x + 35.516
c.
Use your graphing calculator to create a scatter plot and graph of the data on the same set of
axes.
The graphing calculator plot and graph is:
d.
How well does the graph of the best fit parabola fit the data?
The graph of the best fit parabola fits the data very closely.
The table below is the U. S. Census (in millions of people) for the years 1810 through 2000. The x-values
represent the year the Census was taken and the y-values represent the population in millions of
people. Note: x = 0 for the year 1810, x = 10 for the year 1820, etc.
x
y
x
y
1810
0
7.24
1910
100
91.97
1820
10
9.64
1920
110
105.71
1830
20
12.87
1930
120
122.78
1840
30
17.07
1940
130
131.67
1850
40
23.19
1950
140
151.33
1860
1870
50
60
31.44 39.82
1960
1970
150
160
179.32 203.21
1880
70
50.16
1980
170
226.5
1890
1990
80
90
62.95 75.99
1990
2000
180
190
248.71 281.42
a.
Use your graphing calculator to create a scatter plot. Does the data seem to model a quadratic
equation? Explain.
Yes, the data seems to model a quadratic equation because the scatter plot is shaped like the
right half of a parabola
b.
Determine the equation for the parabola of best fit. Round the values of a, b, and c to three
decimal places.
The equation for the best fit parabola is y = 0.007x2 + 0.119x + 7.980
c.
Using your equation of best fit, predict the population for the Census in 2010 and 2020.
The predicted population for 2010 is 311.78 million people and for 2020 is 341.67 million
people
Non–Linear Functions
Lesson 3: Graphing Quadratic Functions and Their Applications
AIIF
Page 57
Practice Exercises
Graph Exercises 1 and 3. Make sure to include the following:
a.
Identify the values of a, b, and c.
b.
Vertex coordinates.
c.
All intercepts.
d.
Line of symmetry.
e.
Several points on either side of the vertex.
NOTE: Round answers to nearest tenth.
1.
y = 5x 2 − 10 x − 1
a.
b.
c.
d.
e.
2.
a = 5; b = –10; c = –1
The coordinates of the vertex are (1, –6)
The y-intercept is (0, –1); the x–intercepts are
approximately (–0.1, 0) and (2.1, 0)
The equation for the line of symmetry is x = 1
Several points on either side of the vertex
include: (0, –1), (2, –1), (–2, 39), (4, 39),
(–4, 119), and (6, 119)
y = −3x 2 + 5x + 12
a.
b.
c.
d.
e.
a = –3; b = 5; c = 12
The coordinates of the vertex are (5/6, 169/12)
The y–intercept is (0, 12); x–intercepts are (3, 0)
and (–4/3, 0)
The equation for the line of symmetry is x = 5/6
Several points on either side of the vertex
include: (2, 10), (–1/3, 10), (5, –38), (–10/3, –38),
(8, –140), and (–19/3, –140)
SJ Page 23
AIIF
Page 58
SJ Page 24
3.
A ball is thrown directly upward from an initial height of 200 feet with an initial velocity of 96 feet per
second. After 3 seconds it will reach a maximum
height of 344 feet. The standard form of a quadratic
equation for a projectile is given
by s(t ) = −16t 2 + v0 t + s0 , where s(t) is the projectiles
height at time t, v0 is the initial velocity, and s0 is the
initial height. What is the equation of the quadratic
function for this problem? What does the y–intercept
represent? Graph the quadratic function. Round
answers to nearest tenth if necessary.
The quadratic equation is s(t ) = −16t 2 + 96t + 200
a.
a = –16; b = 96; c = 200
b.
The coordinates of the vertex are (3, 344)
c.
The y–intercept is (0, 200); the x–intercepts are
approximately (–1.6, 0) and (7.6, 0)
d.
The equation for the line of symmetry us x = 3
e.
Several points on either side of the vertex include: (0, 200), (6, 200), (2, 328), (4, 328), (7, 88),
and (–1, 88)
f.
The y–intercept represents the initial height from where the ball was thrown
NOTE: Vertical scale ratio is 1:14
4.
Suppose that in a monopoly market (a market with a
downward sloping curve) the total cost per week of
producing a particular product is given by the cost
function C ( x ) = 2 x 2 + 100 x + 3600 . The weekly demand
for the product is such that the revenue function is
R( x ) = −2 x 2 + 500 x . Graph both functions on the same
set of axes and shade the region that represents the
area in which the company is making a profit. Find
the points of intersection for the cost and revenue
functions. What do the points of intersection
represent?
The points of intersection are (10, 4800) and (90,
28800). The points of intersection represent the break–
even points.
AIIF
Non–Linear Functions
Lesson 3: Graphing Quadratic Functions and Their Applications
Page 59
SJ Page 25
5.
Determine the quadratic function from the graph at
the right.
The equation of the graph is y = x2 + 8x + 12.
6.
A ball was dropped from a height of approximately 5 feet and a motion detector was used to measure
the time and height of the ball, relative to the ground, as it was falling. The table below is the height, h,
of the ball off the ground in feet after t seconds.
Time t
Height
h
7.
0
4.95
0.04
4.86
0.08
4.73
0.12
4.56
0.16
4.34
0.20
4.08
0.24
3.78
0.28
3.43
0.32
3.04
0.36
2.61
0.40
2.13
a.
Determine the equation for the parabola of best fit.
The equation for the best fit parabola is y = –13.527x2 – 1.632x + 4.949.
b.
How long does it take for the ball to hit the ground? Round your answer to the nearest
hundredth of a second. HINT: Use the quadratic formula.
Sample response: Using the quadratic formula with a = –13.527, b = –1.632, and c = 4.949, the
ball takes approximately 0.55 seconds to hit the ground
You run a bicycle rental business for
tourists during the summer in your
town. You charge $10 per bike and
average 20 rentals a day. An industry
300
journal says that, for every 50–cent
increase in rental price, the average
200
business can expect to lose two
rentals a day. The graph to the right
100
represents the quadratic equation
used to determine how many, if any,
50–cent increases are needed to
maximize revenue. Let x represent
the number of increases to the current charge rate. Negative values for x represent 50–cent decreases.
Use this information and the graph to find the quadratic equation to maximize revenue. What should
you charge per bike rental? What is your maximum profit?
The quadratic equation for the graph is R( x ) = − x 2 − 10x + 200 . You should charge $7.50 per bicycle
rental to have your maximum revenue of $225.
AIIF
Page 60
SJ Page 26
Outcome Sentences
The vertex is determined by
The line of symmetry is used for
Applications of quadratic equations really help me to understand
Quadratic functions and applications of quadratic functions are graphed by
The easiest way for me to write a quadratic equation from a graph is by
The most difficult part of graphing is
Quadratic modeling with the graphing calculator
because
Non–Linear Functions
Lesson 3: Graphing Quadratic Functions and Their Applications
AIIF
Page 61
Teacher Reference
Lesson 3 Quiz Answers
1.
The hang time for the punt is approximately 5 seconds. The maximum height of the punt is 102 feet.
2.
a.
b.
c.
d.
e.
3.
a = 1; b = 10; c = 22
The coordinates of the vertex are (5, –3)
The y–intercept is (0, 22); x–intercepts are
(3.27, 0) and (6.73, 0)
The equation for the line of symmetry is
x=5
Several points on either side of the vertex
include: (4, –2), (6, –2), (2, 6), (8, 6), (0, 22),
and (10, 22)
Answers will vary. A sample response might be: “The best ways to communicate mathematical results
for quadratic functions are through their maximum or minimum value, the vertex, as well as the zeros
(also known as the solutions or x–intercepts)
AIIF
Page 62
Lesson 3 Quiz
1.
Name:
When a football player punts a football, he hopes for a long
“hang time.” Hang time is the total amount of time the ball
stays in the air. A time longer than 4.5 seconds is considered
good. It allows the punting team time to get down the field
and tackle the opponent’s player who will catch the punt. If
a punter kicks the ball with an upward velocity of 80 feet
per second and his foot meets the ball 2 feet off the ground,
the function y = −16t 2 + 80t + 2 represents
the height of the ball y in feet after t
seconds. Sketch the graph of the punt.
What is the maximum height the ball
reaches? What is the hang time of the ball?
Round your answer to the nearest tenth of
a second and nearest tenth of a foot if
necessary.
2.
Graph y = x 2 − 10 x + 22 . Make sure to include the
following:
a.
Identify the values of a, b, and c.
b.
Vertex coordinates.
c.
All intercepts.
d.
Line of symmetry.
e.
Several points on either side of the vertex.
NOTE: Round answers to nearest hundredth.
3.
What are the best ways to communicate mathematical results for quadratic functions in a meaningful
manner?
Non–Linear Functions
Lesson 4: Power Functions
AIIF
Page 63
Lesson 4: Power Functions
Objectives
• Students will understand what constitutes a power function.
• Students will be able to determine if a function is odd, even, or neither.
• Students will be able to graph power functions and their translations.
• Students will be able to solve and graph applications involving power functions
Essential Questions
• Do power functions have patterns that can be used when solving and graphing them?
Tools
• Student Journal
• Setting the Stage transparency
• Activity 1 transparency
• Dry–erase boards, markers, erasers
• Graphing calculator and view screen
Warm Up
• Problems of the Day
Number of Days
• 2 days
Vocabulary
Multiples
Odd function
Power function
Patterns
Even function
AIIF
Page 64
Teacher Reference
Setting the Stage
Before placing the Setting the Stage transparency on the overhead projector, lead a discussion with the class
about patterns. Ask the class about the geometry patterns they have learned about in school. They have also
learned about patterns from graphs of equations and functions. Ask them about some numerical patterns they
have learned about. Tell the class that in the Linear Functions unit they wrote rules for numerical patterns from
tables. Let the class know that algebra looks at the numerical patterns more than the geometrical patterns.
Lead a discussion about multiples. Tell the class that in elementary school they learned that repeated addition
was called multiplication. The students should be familiar with multiples of numbers. Have several students
give the multiples of different numbers such as 2, 3, 4, and 5. The class may have also seen that repeated
multiplication results in a power (you may want to review the parts of a power, base and exponent.) Ask the
class, "How are the multiples of a number related to the values of a power or exponentiation?"
Place the Setting the Stage transparency on the overhead projector. Start with the number 2. Tell the class that
the first multiple of any number is also the first power for that number. So, the first multiple of 2 is also the
first power of 2. The value of the power, 2, will tell us where the next multiple that has the same value as the
next power. That is, the 2nd multiple has the same value of the next power which is 4. Note: The circles relate
which multiples have the same value as the power of 2. The value of 4 tells us the 4th multiple will be where
the multiple values of 2 equals the next power of 2, which is 8. The 8th multiple of 2 is where the next power of
2 and multiple of 2 have the same value, 16. The pattern continues on in this manner. The 16th multiple is 32
which is also the value of the next power of 2; 32nd multiple of 2 is 64 which is where the next power of 2 is
located. This 64 tells us the next multiple where the power of two has the same value as a multiple of two. Ask
the class, "After 32, what is the next multiple of 2 that has the same value as the next power of 2 and what is
that value?" Perhaps the class will say the 64th multiple of 2, has the same value as the next power of 2 which is
128.
Now switch to the number 3 on the transparency. Again, remind the class that the first multiple of 3 will also
be the first power of 3. That value, 3, tells us that the 3rd multiple of 3 will be the next power of 3 which is 9.
The value 9 tells us that the 9th multiple of 3 has the same value as the next power of 3, 27. Have a student
come to the overhead and complete the pattern for the number 3.
Now ask the class to do a similar matching pattern for the number 4 on the blank side of their dry–erase
boards. Give the class five to seven minutes to complete the pattern matching for the number 4. Give the class
graphing calculators if necessary. Have the students hold up their dry–erase boards; walk around to visually
inspect them while a volunteer shares her/his results on the transparency at the overhead.
4
22
2
6
21
1
3
32
31
41
4
1
9
3
9
3
12
4
23
8
15
5
10
2
6
8
4
2
18
6
12
6
5
4
2
1
3
Transparency
Setting the Stage
Non–Linear Functions
Lesson 4: Power Functions
21
7
14
7
24
8
24
16
16
8
33
27
27
9
18
9
30
10
20
10
33
11
22
11
…
…
24
12
34
26
13
84
28
28
14
87
29
30
15
Page 65
AIIF
…
…
25
32
32
16
35
34
17
…
…
36
26
64
36 … 64
18 … 32
8
23
4
22
2
6
21
1
3
32
31
41
4
1
9
3
9
3
12
4
8
2
6
4
4
2
3
2
1
Setting the Stage
Page 66
AIIF
15
5
10
5
18
6
12
6
21
7
14
7
24
8
24
16
16
8
33
27
27
9
18
9
30
10
20
10
33
11
22
11
…
…
24
12
34
26
13
84
28
28
14
87
29
30
15
…
…
25
32
32
16
35
34
17
…
…
36
26
64
36 … 64
18 … 32
Non–Linear Functions
Lesson 4: Power Functions
AIIF
Page 67
Teacher Reference
Activity 1
In this activity, students will create, work with, and solve simple power functions and power equations.
Think aloud as you model the following problem:
The volume of a cube–shaped box is 64 cubic inches. What are the dimensions of the box?
Even though some students might be able to give the answer right away, make sure to model how to properly
set up and solve these types of functions. Some of the concepts you should model are:
•
•
•
Draw a sketch and label the unknown units. For example, because the length, width, and height of a
cube are all equal, we could label each dimension with s.
Write a function, using unknowns representing the problem then substitute the given information. For
example, the function would be V (s ) = s 3 and substituting the volume of 64 would yield the power
equations 64 = s 3 or s 3 = 64 .
Solve the equation. Make sure the solution includes appropriate units. For this example, the solution
would be s = 4 inches.
Have a student model the following problem, on the overhead projector, while the class members follow
along on their dry–erase boards.
A number raised to the fourth power is 81. ( n 4 = 81 )
If the students come up with only one solution, ask them if there is any other number, when raised to the
fourth power, which would equal 81. The goal is for the students to understand that two different numbers,
when raised to the fourth power, would equal 81, namely 3 and –3. Now ask the class when there might be
only one answer to a problem. The understanding here is that the students can differentiate between
problems with just numbers and real-world applications and that sometimes answers just don’t make sense.
If you feel the students need to see another example, have another volunteer model the following problem on
the overhead projector, while the class follows along on their dry–erase boards.
The sum of a number and three, raised to the fifth power is thirty–two.
This problem is a bit more complicated than the previous one. For the class to write the correct equation you
may want to ask questions such as, “What is being raised to the fifth power?” “What must be calculated first,
the sum or the power?” These questions are to help the students get the equation ( x + 3)5 = 32 . These
equations are similar to those in Lesson 1 except they include powers other than squaring. These questions
also help the teachers to see how the students will handle solving power equations of this type. It is helpful to
see the type of strategies the students use to solve an equation such as ( x + 3)5 = 32 , in order to make
adjustments in teaching strategies. Students should be using the same strategies that they have been using to
solve equations in earlier lessons. Make sure the students used their previous equation solving skills and
make any adjustments if necessary.
AIIF
Page 68
Model the following application problem to the class. The class should use their dry–erase boards to solve the
same, or a similar, problem simultaneously. Place Activity 1 transparency on the overhead projector so that
the students can follow along while you read the problem. Ask guiding questions such as, "Which number is
used to replace P in the formula?" and "Which number is used to replace S?"
If P is invested at an interest rate r per year, compounded annually, the future value S at the end of the nth
year is P(1 + r )n . The function that models this is S = P(1 + r )n . What interest rate must a person obtain for a
$10,000 investment to have a future value of $16,288.95 after 10 years?
You may want to model for the students how to calculate the tenth root using their graphing calculators. Let
the students know that they still use the exponent key, ›, but the exponent for a root is the reciprocal of the
original exponent, 1/10. Also inform the students that they should use parentheses for rational exponents
because the calculator uses the order of operations. You could ask, "Is there an undoing operation that will
allow us to undo the power of 10?" The goal is that students understand undoing the operation of a power.
The problem looks like the following:
16288.95 = 10000(1 + r )10
16288.95
= (1 + r )10
10000
10
1.6289 = 10 (1 + r )10
1.05 = 1 + r
0.05 = r
Solution: The interest rate for the investment should be 5%. Ask the students, "Should we consider the
negative root, –1.05? Explain." The students should understand for this type of problem the negative root
would not make sense because we cannot have a negative interest rate.
Another method to determine an nth root with a graphing calculator is to press the  key and then press
the · key to select the 5: x
option. To use this option the students would enter the root index value first,
which is 10 from the example above. Next press the  key followed by the · key to display x
to the
home screen. Next enter the value 1.6289 and press the Í key. The screen shots below display the process
to use this technique for roots on the TI−83 or 84 Plus™ graphing calculator.
Non–Linear Functions
Lesson 4: Power Functions
AIIF
Page 69
Have the students work with their partner on Exercises 1 through 5. Have the students go through the
exercises together and verify answers. When they have finished and are sure of their solutions, have them
group with another pair to compare their answers. After the students have verified their answers with
another pair, have student volunteers share their answers with the class. They can write their solution on the
overhead or on the board. While the students are working, circulate to ask guiding questions and provide
encouragement. You might want to model Exercise 5 with the class or have a student model this exercise.
AIIF
Page 70
SJ Page 27
Activity 1
In this activity, you will write and solve power type equations and their applications. Let's look at the
following application problem:
The area of a cube–shaped box is 64 cubic inches. What are the dimensions of the box?
What are the steps necessary to setup and solve problems of this type? We need to start
by labeling the known and unknown (variable) information. Next, we need to write an
equation with a single variable from the given information. Then, we need to solve the equation and answer
the original question or questions.
For Exercises 1 through 3, write and solve a power type equation.
1.
The cube of a number is 125. What is the number?
The equation is n 3 = 125 . Solving this equation we get that the number is 5; n = 5.
2.
Six is added to a number that was raised to the sixth power. If the sum is 735, what was the number
that was raised to the sixth power?
The equation is n 6 + 6 = 735 . Solving this equation we get that the number is 3 or –3; n = 3 or n=–3.
3.
The difference of a number and six, raised to the fourth power, is 256. What are the numbers?
4
The power equation is ( n − 6 ) = 256 . Solving this equation we get negative 2 or positive 10:
4
( n − 6)
4
= 4 256
n − 6 = ±4
n = ±4 + 6
n = 4 + 6 = 10
or
n = −4 + 6
n = −2
4.
The volume of a spherical weather balloon is 523.3 cubic meters. What is the diameter of the weather
4
balloon? NOTE: The formula for the volume of a sphere is V = π r 3 where r is the radius. Use 3.14 for
3
the value of π .
Solving this equation we get that the diameter of the weather balloon is 10 feet.
Non–Linear Functions
Lesson 4: Power Functions
AIIF
Page 71
SJ Page 27 (cont.)
4
523.3 = (3.14)r 3
3
⎛ 3⎞
⎛ 3 ⎞⎛ 4 ⎞
3
⎜ ⎟ 523.3 = ⎜ ⎟ ⎜ ⎟ (3.14)r
4
4
3
⎝ ⎠
⎝ ⎠⎝ ⎠
392.475 = 3.14r 3
392.475 3.14 r 3
=
3.14
3.14
3
5.
392.475 3 3
= r
3.14
5≈r
A couple plans to invest $25,000.00 into an account that is compounded annually for 25 years. They
hope to have $75,135.86 after the 25 years. What interest rate will guarantee that their investment of
$25,000.00 will grow to $75,135.86 after the 25 years? NOTE: S = P(1 + r)t, where S is the value of the
investment, P is the amount invested, r is the interest rate (as a decimal), and t is the number of years
invested.
The interest rate they need is r = 4.5%.
75,135.86 = 25,000(1 + r ) 25
75,135.86 ⎛ 25,000 ⎞
=⎜
(1 + r ) 25
⎜ 25,000 ⎟⎟
25,000
⎝
⎠
25
3.0054 = (1 + r )
25
3.0054 = 25 (1 + r ) 25
1.045 = 1 + r
1.045 − 1 = 1 − 1 + r
0.045 = r
AIIF
Page 72
Activity 1
Transparency
If P is invested at an interest rate r per year,
compounded annually, the future value S at the
end of the nth year is P(1 + r )n . The function that
models this is S = P(1 + r )n . What interest rate must
a person obtain for a $10,000 investment to have a
future value of $16,288.95 after 10 years?
Non–Linear Functions
Lesson 4: Power Functions
AIIF
Page 73
Teacher Reference
Activity 2
In this activity, students will investigate the graphs of power functions using the classroom graphing
calculator. Model how to graph y = x 2 and y = x 4 at the same time on the graphing calculator while the
students graph them on their calculators. The class should be familiar with the first equation. Use the
TRACE key to switch between the graphs so the class can determine which graph represents which equation.
You may want to have a list of questions prepared to ask the students after the graphs are displayed on the
calculator view screen and on the students' calculators. Here are some questions:
•
•
•
•
•
What is the shape of the graph?
Does the graph have a minimum or maximum value?
What do you notice about the portion of the graph to the left of the y–axis compared to the portion to
the right of the y–axis?
What are the similarities and differences between the two graphs?
What might be the best way to determine which graph is which without using a table of values or a
TRACE key?
The graphs below are of the functions y = x 2 , y = x 4 , and y = x 6 .
The graphs below are of the functions y = x 3 , y = x 5 , and y = x 7 .
Have a student volunteer record the responses on the board while the students record the responses in their
student journals.
AIIF
Page 74
Next, have a volunteer model graphing the two equations y = x 3 and y = x 5 on a graphing calculator. Have the
student, along with the class list the similarities and differences between these equations. Use the TRACE
key to switch between the graphs so the class can determine which graph represents which equation. You
may want to have questions prepared to ask after the graphs are displayed on the graphing calculator view
screen and on the students' graphing calculators. Here are some questions:
•
•
•
•
•
What is the shape of the graph?
Does the graph have a minimum or maximum value?
Is there symmetry of any type?
What are the similarities and differences between the two graphs?
What might be the best way to determine which graph is which without using a table of values or a
TRACE key?
Have the class complete Part A of Activity 2 in their journals in their pairs. Have student pairs share their
results with the class.
Students could also investigate, using the TRACE key, multiplying by a coefficient other than 1 and
comparing it to multiplying by a coefficient of 1. For example, Y1 could contain y = x3 and Y2 could
contain y = 3x 3 . Students could predict what they expect the results to be before graphing the equations or
looking at the table of values for the equations. Now, have the class complete Part B of Activity 2 in their
journals in their pairs. Have student pairs share their results with the class.
Power functions of the type y = x n , where n is an even integer greater than or equal to 2, are called even
functions because the left half of the graph is a mirror image of the right half (vertical symmetry). They are
also called even functions because you obtain the same y values for both positive and negative values of x. For
example, if y = 4 when x = 1, then y = 4 when x = –1. In function notation, a function is an even function when
f (x) = f (–x). Power functions of the type y = x n , where n is an odd integer greater than or equal to 3, are called
odd functions because the left half of the graph is a mirror image of the right half only reflected about the x–
axis. They are also called odd functions because you obtain the opposite y values for negative values of x that
you obtained for positive values of x. For example, if y = 4 when x = 1, then y = –4 when x = –1. In function
notation, a function is an odd function when –f (x) = f (–x). It is advised that students look at symmetry
and/or table values to determine if the function is an even function or an odd function. Model using the
graphing calculator how the students could enter the function into the Y= and then check the table values to
determine when the function is even or odd. Students may think that they only have to look at the exponent
to determine if the function is even or odd. Although even functions will have an even exponent and odd
functions will have an odd exponent, if the function has any translation from the origin the function will no
longer be an even or odd function since it will not adhere to the rules to be an even or odd function.
Now, we will investigate horizontal and vertical translations for power functions. Have the class use their
graphing calculators and input the equation y = x 2 into Y1 (make sure to have them clear out any existing
equations in the Y= editor.) Tell the class to add any number between –5 and +5 after x 2 . Have students share
their results and any conclusion from the number they added. You could have the students who are sharing
their results, use the calculator view screen to display their graph while they share their results and any
conclusions. Have the class work on Part C in Activity 2 and have students share their findings with the class.
AIIF
Non–Linear Functions
Lesson 4: Power Functions
Page 75
Use the same TRACE process to have students add a number between –5 and +5 to the x value before
squaring. For example, use ( x + 2 ) . You may want to show the class how to do this by pressing the following
2
TI–83 or 84 Plus™ (please modify the steps for your graphing calculator) key sequence £ „ Ã Á (or any
other number) ¤ ¡. Their equation should resemble something like y = ( x + 2 ) . Again, have students share
2
their results and any conclusion from the number they added. You could have the students who are sharing
their results, use the calculator view screen to display their graph.
From the two previous investigations, the class should understand that adding a value after x 2 results in a
vertical translation and adding a value with x before squaring results in a horizontal translation. Ask the class,
"What results would we expect if we put the two translations together? Can you give an example to support
your explanation?" Students should be able to give examples of the two translations in a single equation, such
as y = ( x + 3 ) + 2 . Ask the class, "Would changing the exponent, from two to three, affect the translations?
2
Explain." The class should realize that the translations would be unaffected by a change in exponents. Have
the class work on Part D in Activity 2 and have students share their findings with the class.
Have the students work in pairs on Exercises 1 through 10. Have students share their results with the class.
Have individuals or student pairs check their results with other students or student pairs. Lead a class
discussion on the characteristics and graphs of power or power–like functions. Include guiding questions
such as:
• What are the major differences between odd and even functions?
• How can the differences between odd and even functions be used to identify their graphs?
• What could be the best method to determine the equation of a power function from its graph?
• How do translations affect the graph of power functions?
• How does multiplying a power function by a constant affect its graph?
Note: For Practice Exercise 10, you may want to discuss or model this problem before assigning it to the class.
Lead a discussion with the class about the techniques they could use to solve the problem. Have a student list
the techniques on the board and the class could write these techniques and ideas down in their student
journal by the exercise to have when they are trying to solve the problem.
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Page 76
SJ Page 28
Activity 2
In this activity, you will investigate the graphs of power and power–like functions. In your descriptions
include whether the graph is an even or odd function.
Part A: How do different powers affect the graph of y = x n ?
Function
y=x
n
Describe or Draw
General Shape
Describe location of
maximum or
minimum
Describe similarity
or difference to the
graph of y = x 2
y = x3
See Teacher Reference
for the general shape
of the graph
No maximum or
minimum
The graph is not
similar at all to y =
x2. This is the graph of
an odd function
y = x4
Parabolic in shape
The minimum value is
at (0, 0)
The graph is very
similar to the graph
of y = x2
y = x5
See Teacher Reference
for the general shape
of the graph
No maximum or
minimum
The graph is not
similar at all to y =
x2. This is the graph of
an odd function
y = x6
Parabolic in shape
The minimum value is
at (0, 0)
The graph is very
similar to the graph
of y = x2
y = x7
See Teacher Reference
for the general shape
of the graph
No maximum or
minimum
The graph is not
similar at all to y =
x2. This is the graph of
an odd function
Write your overall conclusion as to how different powers affect the graph of y = x n .
The answers will vary. A sample response might be: "As the value of n increases the graph of y = xn becomes
more condensed and compact. The shapes of even powers of x stay roughly the same and the shapes of odd
powers of x stay roughly the same as well."
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Non–Linear Functions
Lesson 4: Power Functions
Page 77
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n
Part B: How do different coefficients affect the graph of y = ax ?
Function
y = ax
n
Describe or Draw
General Shape
Describe location of
maximum or minimum
Describe similarity or
difference to the graphs
of y = x 2 or y = x 3
y = −x 2
Parabolic in shape but
reflected about the x–axis
Maximum value at (0, 0)
y = 4x 2
Parabolic in shape but
grows at a faster rate
than y = x2
Minimum value at (0, 0)
y = −9 x 2
Parabolic in shape but
reflected about the x–axis
Maximum value at (0, 0)
−1 2
x
2
Parabolic in shape but
reflected about the x–axis
and decreasing at a
slower rate than y = x2
Maximum value at (0, 0)
y = −x 3
Same shape as y = x3 but
reflected about the x–axis
No maximum or
minimum
y = −5x 3
Same shape as y = x3 but
reflected about the x–axis
and decreasing at a faster
rate than y = x3
No maximum or
minimum
y = 5x 3
Same shape as y = x3 but
increasing at a faster rate
than y = x3
No maximum or
minimum
The graph is similar to
the graph of y = x3 but
increases at a faster rate
Same shape as y = x3 but
increasing at a slower
rate than y = x3
No maximum or
minimum
The graph is similar to
the graph of y = x3 but
increases at a slower
rate
y=
y=
1 3
x
2
The graph is similar to
the graph of y = x2 but is
reflected about the x–
axis
The graph is similar to
the graph of y = x2 but
increases at a faster rate
The graph is reflected
about the x–axis and
decreases at a faster
rate
The graph is reflected
about the x–axis and
decreases at a slower
rate
The graph is similar to
the graph of y = x3 but is
reflected about the y–
axis
The graph is reflected
about the y–axis and
decreases at a faster
rate
Write your overall conclusion as to how different coefficients affect the graph of y = ax n . The answers will
vary. A sample response might be: "A coefficient greater than 1 causes the graph to increase at a faster rate. A
negative coefficient causes the graph to be reflected about the x–axis."
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Part C: How does adding or subtracting a constant, k, to y = x n affect the graph of the equation?
Function
n
y=x ±k
Describe or Draw
General Shape
Describe location of
maximum or minimum
Describe similarity or
difference to the graphs
of y = x 2 or y = x 3
y = x2 + 1
Parabolic in shape
but vertically
translated by 1 unit
from the origin
Minimum value at (0, 1)
The graph is similar to
the graph of y = x2 but is
vertically translated
from the origin
y = x2 + 3
Parabolic in shape
but vertically
translated by 3 units
from the origin
Minimum value at (0, 3)
The graph is similar to
the graph of y = x2 but is
vertically translated
from the origin
y = x2 − 2
Parabolic in shape
but vertically
translated by –2 units
from the origin
Minimum value at (0, –2)
The graph is similar to
the graph of y = x2 but is
vertically translated
from the origin
y = x2 − 4
Parabolic in shape
but vertically
translated by –4 units
from the origin
Minimum value at (0, –4)
The graph is similar to
the graph of y = x2 but is
vertically translated
from the origin
y = x3 + 5
Same shape as y = x3
but vertically
translated by 5 units
from the origin
No maximum or
minimum
The graph is similar to
the graph of y = x3 but is
vertically translated
from the origin
y = x3 + 7
Same shape as y = x3
but vertically
translated by 7 units
from the origin
No maximum or
minimum
The graph is similar to
the graph of y = x3 but is
vertically translated
from the origin
y = x3 − 6
Same shape as y = x3
but vertically
translated by –6 units
from the origin
No maximum or
minimum
The graph is similar to
the graph of y = x3 but is
vertically translated
from the origin
Write your overall conclusion as to how adding or subtracting different constants affect the graph of y = x n .
The answers will vary. A sample response might be: "The graphs are the same as y = x2 or y = x3 but the
graphs have been vertically translated."
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Non–Linear Functions
Lesson 4: Power Functions
Page 79
SJ Page 31
Part D: How does adding or subtracting a constant, h, to the x–value before completing the power, in the
equation y = x n , affect the graph?
Function
n
y = (x ± h)
y = ( x + 1)2
Describe or Draw
General Shape
Describe location of
maximum or minimum
Describe similarity or
difference to the graphs
of y = x 2 or y = x 3
Parabolic in shape
but horizontally
translated by –1 unit
from the origin
Minimum value at (–1, 0)
The graph is similar to
the graph of y = x2 but is
horizontally translated
from the origin
y =(x + 3)
2
Parabolic in shape
but horizontally
translated by –3 units
from the origin
Minimum value at (–3, 0)
The graph is similar to
the graph of y = x2 but is
horizontally translated
from the origin
y =(x−2)
2
Parabolic in shape
but horizontally
translated by 2 units
from the origin
Minimum value at (2, 0)
The graph is similar to
the graph of y = x2 but is
horizontally translated
from the origin
y =(x+4)
3
Same shape as y = x3
but horizontally
translated by –4 units
from the origin
No maximum or minimum
The graph is similar to
the graph of y = x3 but is
horizontally translated
from the origin
y =(x+6)
3
Same shape as y = x3
but horizontally
translated by –6 units
from the origin
No maximum or minimum
The graph is similar to
the graph of y = x3 but is
horizontally translated
from the origin
y =(x −3)
3
Same shape as y = x3
but horizontally
translated by 3 units
from the origin
No maximum or minimum
The graph is similar to
the graph of y = x3 but is
horizontally translated
from the origin
Write your overall conclusion of how adding or subtracting a constant to the x–value in the equation y = x n
affected the graph.
The answers will vary. A sample response might be: "The graphs are the same as y = x2 or y = x3 but the
graphs have horizontally translated."
Write your overall conclusion as to what affect the values of a, h, k, and n have on the graph
of y = a( x ± h )n ± k . The answers will vary.
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SJ Page 32
For Exercises 1 through 4, determine the following:
a.
Determine if the graph represents a power or power–like function or not.
b.
Determine if the graph has a maximum or minimum value. If it does, state the value of the
maximum or minimum.
c.
If the function represented by the graph is a power function, determine if it is even, odd, or
neither.
1.
2.
a.
b.
c.
It is a power–like function
Has a minimum value of –9
The function is neither even nor odd
3.
a.
b.
c.
Is a power–like function
Has a maximum value of 8
The function is neither even
nor odd
a.
b.
Is a power function
There is no maximum nor
minimum
It is an odd function
4.
a.
b.
c.
Is not a power function
Seems to have a minimum value
approaching 0
This is neither an even nor odd function
c.
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Non–Linear Functions
Lesson 4: Power Functions
Page 81
SJ Page 33
5.
Match the equation with its graph.
a.
y = 2x4
This equation matches graph B.
b.
y = x3
This equation matches graph C.
c.
y = –x3
This equation matches graph A.
B.
A.
C.
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Page 82
SJ Page 34
For Exercises 6 through 10, state any vertical or horizontal translation from the first equation to the second.
Sketch a rough graph of the equations showing translation
(do not worry about scale).
6.
y = x3 and y = x3 + 4.
There is a vertical translation of +4 units from the origin.
7.
y = x4 and y = x4 – 3.
There is a vertical translation of –3 units from the origin.
8.
y = x5 and y = (x – 6)5.
There is a horizontal translation of +6 units from the
origin.
9.
y = x5 and y = (x – 6)5 – 2.
There is a vertical translation of –2 units and a
horizontal translation of +6 units from the origin.
10.
y = x6 and y = (x + 1)6 + 5.
There is a vertical translation of +5 from the origin and a
horizontal translation of –1 from the origin.
AIIF
Non–Linear Functions
Lesson 4: Power Functions
Page 83
SJ Page 35
Practice Exercises
For the Exercises 1 through 3, write and solve a power–like equation.
1.
The fifth power of a number is 243. What is the number?
The power equation is n5 = 243. Solving this equation we get the number 3; n = 3.
2.
Nine is subtracted from a number that is raised to the seventh power. If the difference is 119, what
was the number that was raised to the seventh power?
The power equation is n7 – 9 = 119. Solving this equation we get the number 2; n = 2.
3.
The sum of a number and three, raised to the third power, is 1,331. What is the number?
The power equation is (n + 3)3 = 1331. Solving this equation we get 8:
3
( n + 3)3
= 3 1331
n + 3 = 11
n + 3 − 3 = 11 − 3
n=8
4.
The volume of a cubic box is approximately 1521 cubic inches. What are the lengths of the sides of the
cubic box? Round your answer to the nearest tenth of an inch.
The volume formula for a cube box is V = s3. Solving this equation we get that the length of the sides
of the box is approximately 11.5 inches.
5.
Darnell and Shanice plan to invest $50,000.00 into an account that is compounded annually at a rate of
3.5%. Create a table of values that represents what their investment is worth after 4, 8, 12, and 16 years.
NOTE: S = P(1 + r)t, where S is the value of the investment, P is the amount invested, r is the interest
rate (as a decimal), and t is the number of years invested. Round the value of the investment to the
nearest cent.
Years Invested (t)
4
8
12
16
6.
Value of Investment in dollars (S)
57,376.15
65,840.45
75,553.43
86,699.30
Darnell and Shanice plan to use the total value of the investment in 16 years for a college education for
their only child. Approximately how much will they have available each year, for four years, for their
child's education? Round your answer to the nearest thousand dollars.
The couple will have about $21,675 per year for their child's college education.
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Page 84
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For Exercises 7 through 9, complete the following:
a.
State if it is a power or power–like function or not.
b.
State if it has a maximum or minimum value and state the value of the maximum or minimum.
c.
State if it the function is even, odd, or neither.
d.
State any vertical or horizontal translation from the origin.
e.
Sketch a rough graph of the power or power–like function.
7.
y = –x4
a.
b.
c.
d.
8.
y = (x + 3)3
a.
b.
c.
d.
9.
The equation is a power function.
The function has a maximum value of 0 at x = 0.
The function is an even function.
There are no translations from the origin.
The equation is a power–like function.
The function has no maximum or minimum
values.
The function is an odd function.
There is a horizontal translation of –3 from the
origin.
y = (x – 2)5 – 4
a.
b.
c.
d.
The equation is a power–like function.
The function has no maximum or minimum
values.
The function is an odd function.
There is a horizontal translation of +2 and a
vertical translation of –4 from the origin.
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Non–Linear Functions
Lesson 4: Power Functions
Page 85
SJ Page 37
10.
Determine the power–like function from the given graph.
The equation for the power function is y = (x – 2)3.
(5, 27)
25
(2, 0)
(–1, –27)
–25
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Page 86
SJ Page 38
Outcome Sentences
A power function is
The difference between an even and an odd function is
Applications of power functions really help me to understand
When graphing power functions
Vertical and horizontal translations from the origin are
The most difficult part of power functions is
AIIF
Non–Linear Functions
Lesson 4: Power Functions
Page 87
Teacher Reference
Lesson 4 Quiz Answers
1a.
n4 = 256; the number is 4 or –4.
1b.
(n – 3)5 = 243; the number is 6.
2.
The power–like function has a horizontal translation of 1 unit from the origin and a vertical
translation of 2 units from the origin. The coordinates of the minimum value are (1, 2).
3.
a.
b.
c.
d.
e.
The graph represents a power–like function.
The graph has no minimum or maximum value.
The function is neither even or odd because it has been translated horizontally and vertically
from the origin.
There is a horizontal translation of –2 units from the origin and a vertical translation of a –3
units from the origin.
3
The equation is y = ( x + 2 ) − 3
AIIF
Page 88
Lesson 4 Quiz
1.
2.
Name:
Write and solve a power type equation for the following:
a.
A number raised to the fourth power is 256. What is the number?
b.
The difference of a number and 3 raised to the fifth power is 243. What is the number?
4
For the given power–like function, y = ( x − 1 ) + 2 , state
any vertical and/or horizontal translation from the
origin, state the coordinates of the minimum or
maximum value if there is one, and sketch a rough graph
of the power–like function.
3.
For the given graph:
a.
State if the graph represents a power or
power–like function or not.
b.
Determine if the graph has a maximum or
minimum value and state the value of the
maximum or minimum.
c.
Determine if the graph represents an even
function, odd function, or neither.
d.
20
10
–10
State any vertical or horizontal translation
from the origin.
–20
e.
Write the equation for the graph
Note: Units on x–axis are scaled 1:1
Non–Linear Functions
Lesson 5: Inverse Variation
AIIF
Page 89
Lesson 5: Inverse Variation
Objectives
• Students will be able to write equations involving direct variation applications
• Students will be able to calculate the constant of proportionality k
• Students will be able to write equations involving inverse variation applications
• Students will be able to graph direct variation equations
• Students will be able to graph inverse variation equations
• Students will be able to identify inverse variation phrases
Essential Questions
• How is inverse variation used in real–world application problems?
• How is direction variation used in real–world application problems?
Tools
• Student Journal
• Setting the Stage transparency
• Dry–erase boards, markers, erasers
• Graphing calculator and view screen
• Construction paper
Warm Up
• Problems of the Day
Number of Days
• 2 days
Vocabulary
Direct
Inverse variation
Inverse
Constant of proportionality
Direct variation
AIIF
Page 90
Teacher Reference
Setting the Stage
Place the Setting the Stage transparency on the overhead projector. Cover the bottom part displaying the first
two lines of text containing the phrases "Direct Variation Inverse Variation", and "Varies Directly Varies
Inversely." Lead a discussion about Direct and Inverse. Ask the class, “What is the difference between direct
and inverse? Can you give examples of each?” Have the class work in groups of four and give them 2 minutes
to come up with a list of differences between direct and inverse. Have groups share their list with the rest of the
class. Have a volunteer list the class responses on the board or on the overhead projector.
Now uncover the rest of the Setting the Stage transparency. Have volunteers read each statement. Have the
students continue working in their groups and have them discuss the similarities and differences between the
statements. Tell the students to determine the two variables in each statement and which one is the
independent variable and which is the dependent variable. This will get the students to think about their
understanding of variables discussed in the Linear Functions unit. Have different groups share their results for
at least one of the statements. The key concept is that the students recognize that for inverse variation, the
dependent variable decreases proportionately as the independent variable increases.
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Non–Linear Functions
Lesson 5: Inverse Variation
Page 91
Setting the Stage
Transparency
Direct Variation
Inverse Variation
Varies Directly
Varies Inversely
The number of cricket chirps
increases as the temperature
increases.
The amount of gasoline
decreases as the miles driven
increases.
The amount of money earned
increases as the number of
hours worked increases.
The temperature of hot cocoa
decreases as the amount of
time increases.
School grades increase as the
number of hours spent
studying increases.
The amount of available cell
phone minutes decreases as
the amount of time we use our
cell phone increases.
Miles driven increase as the
time spent driving increases.
The amount of available
energy we have decreases as
the amount of time exercising
increases.
Amount of confidence
increases as our grades
increase.
Amount of available money
decreases as the number of
items purchased increases.
AIIF
Page 92
Teacher Reference
Activity 1
In this activity, students will create a bar graph that represents the equation y = 1/x, for x values 1 through 10.
Have the students work individually or in pairs. Tell the class to cut out the grid template and the strip
cutouts. Point out the 1–unit location on the vertical axis on the grid. The 1–unit value represents the length of
one of the strips. Have the class measure one of the strips in millimeters. The 1–unit value represents 180
millimeters (mm). Tell the class to place the 1–unit strip above the 1 on the horizontal axis. Have the class cut
the remaining strips so that their lengths represent the fractions 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, and
1/10 and place them along the horizontal axis above the values for 2 through 10. For 1/7, tell the class to
round to the nearest millimeter, 26. For 1/8, tell the class to try to cut half way between 22 and 23 millimeters
to get the 22.5 length. Provide each student or student pair with scotch tape to tape down each strip.
Have the class write the fraction values above each strip on the grid and then calculate the decimal value for
the fraction to the nearest hundredth. Students should also record these fraction and decimal values in the
table provided in their student journal. A sample of what the bar graph should look like is displayed below.
y
1
1 unit
1/2
1/3
1/4
1/5
1
2
3
4
5
1/6
1/7
6
7
1/8
1/9
8
9 10
1/10
x
AIIF
Non–Linear Functions
Lesson 5: Inverse Variation
Page 93
Activity 1
SJ Page 39
In this activity, you will create a bar graph that represents the equation y = 1/x, for x values 1 through 10.
1.
Cut out the grid template. Obtain a piece of construction paper from your teacher and cut a strip that is 1
centimeter wide by 180 millimeters long. Notice the 1–unit location on the vertical axis of the grid. The 1–
unit value represents the length of one of the strip cut to 180 millimeters.
2.
Place the 1–unit strip at unit 1 on the x–axis. Cut the remaining strips so that their lengths represent the
fractions 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, and 1/10 the length of the unit strip cut in Exercise 1.
Place the cut strips along the horizontal axis at the values for 2 through 10. Tape down each strip.
3.
Write the fraction values above each strip on the grid and
then calculate their decimal values to the nearest hundredth.
Record the fraction and decimal values in the table to the
right.
4.
Use the grid below to create a scatter plot of the x-values
from the table and then draw a smooth curve connecting the
points onyyour scatter plot.
x
x
1
2
3
4
5
6
7
8
9
10
y = 1/x
1/1
1/2
1/3
1/4
1/5
1/6
1/7
1/8
1/9
1/10
Decimal Value
1.00
0.50
0.33
0.25
0.20
0.17
0.14
0.13
0.11
0.10
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Page 94
SJ Page 40
5.
What do you notice about the values of y, which represent the lengths of the strips, as the values of x
increase?
The values of y decrease as the values of x increase.
6.
As the values of x get larger and larger, what value does y seem to approach?
As the values of x get larger, the values of y get closer and closer to 0.
7.
Using your graphing calculator, determine the values of y for each x–value in the table below. Write the
y–values in the right column of the table.
x
1/2
1/5
1/10
1/50
1/100
1/500
1/1000
1/50000
1/100000
1/1000000
y = 1/x
2
5
10
50
100
500
1000
50,000
100,000
1,000,000
8.
From your results from Exercise 7, as x gets closer to 0 what value does y get closer to?
As x gets closer to 0, the values of y get closer to positive infinity.
9.
Can we find the value of y for x = 0? Explain.
We cannot find the value of y for x = 0 because we cannot divide by 0, the result is undefined.
10.
Investigate variations of the inverse function by using your graphing calculator.
a.
Graph y = 2/x. Describe the differences between this graph and the graph y = 1/x.
Answers will vary. A sample response might be: The graphs are similar. The values of y = 2/x are
twice as large as the values for y = 1/x.
b.
Graph y = 3/x. Describe the differences between this graph and the graph y = 1/x.
Answers will vary. A sample response might be: The graphs are similar. The values of y = 3/x are
three times larger than the values for y = 1/x.
c.
Graph y = –1/x. Describe the differences between this graph and the graph y = 1/x.
Answers will vary. A sample response might be: The graph of –1/x is in quadrants two and four
compared to one and three for y = 1/x. The values are also opposites.
d.
Graph y = –2/x. Describe the differences between this graph and the graph y = 1/x.
Answers will vary. A sample response might be: The graph of –2/x is in quadrants two and four
compared to one and three for y = 1/x. The values of –2/x are opposite the values of 1/x and twice
as large.
AIIF
Non–Linear Functions
Lesson 5: Inverse Variation
Page 95
SJ Page 41
Activity 1 Grid Template
y
Cut Here
1 unit
1
2
3
4
5
6
7
8
9
10
x
AIIF
Page 96
Cut Here
Activity 1 Strip Cutouts
SJ Page 43
Non–Linear Functions
Lesson 5: Inverse Variation
AIIF
Page 97
Teacher Reference
Activity 2
In this activity, students will investigate real–world applications of inverse variation. Tell the class that in the
last activity, they looked at the equation y = 1/x and discovered that as x increased y decreased. The class
should also realize from the last activity and Setting the Stage that as x increased y decreased proportionately
and so the equation y = 1/x represents inverse variation. From this we can conclude that the converse is also
true, as x decreases then y increases proportionally.
Write the general inverse variation equation, y = k/x on the board or a blank transparency on the overhead
projector. Write the words for the equation, "y varies inversely with x" and "y varies inversely proportional to
x." Tell the class that k represents the constant of proportionality. Let the class know that inversely is the key
word here. Another way of looking at the equation is xy = k, which says that the product is k, and is known as
the constant of proportionality.
Have the students investigate inverse variation and k by using the xy = k equation by doing Exercise 1. The
students could work individually or in pairs. Have a volunteer model a similar problem, such as xy = 36, on
the board or a blank transparency at the overhead. The class could discuss the answers for parts a through f
together as a class. While the class is working, circulate to ask guiding questions and provide encouragement.
Now, let's look at real–world applications of inverse variation. Let the class know that k must be calculated
from information given in the problem. Model the following problem with the class to find k:
The number of hours, h, it takes for a block of ice to melt varies inversely with the temperature, t. If it
takes 2 hours for a square inch of ice to melt at 65° F, how long will it take for the ice to melt at 60° F?
Have a volunteer come to the board or overhead projector and model solving this problem with the class
assisting. Ask guiding question as needed.
• The first question is, "How do the variables in the problem match with the variables in the general
inverse variation equation y = k/x?" The class should make the connection that since h will be
increasing as the temperature decreases, h represents y and then t must represent x. Have the
volunteer write the inverse variation equation, with class assistance if necessary, for the given
variables in the problem. The student volunteer should write h = k/t.
• Ask the class, "What do we know from the problem and what don't we know?" The class should
state that they know it takes 2 hours for the ice to melt at 65° F, but they don't know how long it will
take to melt at 60° F and they don't know the constant of proportionality k.
• Ask the class, "Do we solve for k first or do we solve for the time it takes the ice to melt at 60° F?" The
class should realize that they can't solve for the time it takes the ice to melt at 60° F until they know
the constant of proportionality k. Have the student volunteer solve for k with the class assisting as
necessary. The result is that k = 2(65) = 130. After the student volunteer has solved for k, have
her/him answer the original question in the problem by finding the time it will take the ice to melt
at 60° F. Namely, h = 130/60 ≈ 2.17 hours.
Have a second and third volunteer model solving a problem while the class parallels with a similar, but
different problem:
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Page 98
Volunteer:
Class:
Volunteer:
Class:
y varies inversely with x. y = 7 when x = 6. Use the equation y = k/x. What is the value of
y when x = 21? The volunteer should get that k = 42. Similarly we find that y = 2 when
x = 21.
y varies inversely with x. y = 12 when x = 2. What is the value of y when x = 6? The class
should get that k = 24. The class should find that y = 4 when x = 6.
y varies inversely with the square of x. y = 18 when x = 2. Use the equation y = k/x2. What
is the value of y when x = 3? The volunteer should get that k = 72 and that y = 8 when
x = 3.
y varies inversely with the square of x. y = 6 when x = 4. What is the value of y when
x = 8? The class should get that k = 96 and that y = 1.5 when x = 8.
Discuss with the class the difference between the last two problems they solved. The key idea here is the x was
squared in the last problem. Let the class know that y could also vary inversely with the cube of x, x3, or y
could vary inversely with the fourth power of x, x4.
Have students work in pairs on Exercises 2 through 8. For Exercise 8, you might want to lead a short
discussion and ask the class, "What sort of information in the graph can be used to write the inverse variation
equation?" The class should realize that they have several ordered pairs in the graph which could be used to
find the constant of proportionality and then use the value of k to write the inverse variation equation. Have
student pairs check their results with other student pairs. Bring the class together and have volunteers share
their results for any problems that the class had trouble with.
Non–Linear Functions
Lesson 5: Inverse Variation
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Activity 2
SJ Page 45
In this activity, you will solve inverse variation problems and real–world inverse variation problems. The
general form for an inverse variation equation is y = k/x, where k is called the constant of proportionality.
Another way of writing this equation is xy = k. You will investigate inverse variation and k by using the xy = k
equation.
1.
Find five different sets of values (ordered pairs) that make xy = 24 true.
Answers will vary. A sample response might be: Five different sets of ordered pairs are (1, 24), (2, 12), (4,
6), (–2, –12), and (–8, –3)
a.
As the values of x increase what do you notice about the values of y?
The answers may vary. A sample response might be: As the values of x increased the values of y
had to decrease.
b.
Why would the values of y have to decrease as x increases to keep the equation true?
The answers may vary. A sample response might be: If both the values of y and x increased then
we wouldn't be able to keep the constant value of 24.
c.
If the x–value doubles what happens to the y–value?
The answers may vary. A sample response might be: If the value of x doubled then the value of y
would be half as much as it was previously.
d.
If the x–value triples what happens to the y–value?
The answers may vary. A sample response might be: If the value of x triples then the value of y
would have to be one–third as much as it was previously.
e.
What happens to the relationship between x and y if we change the constant to a different
number such as 36?
The answers may vary. A sample response might be: Both x and y must adjust so that the
product is now 36.
f.
Why do you think equations in the form of xy =k, where k is constant, are called inverse variation
equations?
The answers may vary. A sample response might be: Because as one variable increases the other
must do the opposite or the "inverse" of increasing which would be decreasing.
For Exercises 2 and 3, use the given information to solve for the constant of proportionality k and then for the
unknown value of y.
2.
If y varies inversely with x and y = 34 when x = 1/68, what is the value of y when x = 2?
The value of k is 0.5. Therefore, y = 0.25 when x = 2.
3.
If y varies inversely with the cube of x and y = 10 when x = 4, what is the value of y when x = 2?
The value of k is 640. Therefore, y = 80 when x = 2.
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For Exercises 4 through 7, use the information given in the problem to find the constant of proportionality k
and answer the question.
4.
5.
6.
The number of hours, h, it takes to mow a lawn varies inversely with the
number of people mowing the lawn at the same time.
a.
If it takes 3 hours for 3 people to mow the lawn, how long will it take
5 people to mow the same lawn?
The constant of proportionality k = 3(3) = 9. It will take 5 people 1.8 hours to mow the lawn, h =
9/5.
b.
Write an inverse variation equation for the problem.
The inverse variation equation is h = 9/p, where p represents the number of people mowing the
lawn.
Boyle's law states that in a perfect gas where mass and temperature are kept
constant, the volume, V, of the gas will vary inversely with the pressure, P. A
volume of gas, 550 centimeters cubed, is under a pressure of 1.78 atmospheres.
a.
If the pressure is increased to 2.5 atmospheres, what is the volume of the
gas?
The constant of proportionality k = 550(1.78) = 979. The volume of the gas is
391.6 centimeters cubed, V = 979/2.5.
b.
Write an inverse variation equation for Boyle's law.
The inverse variation equation for Boyle's law is V = 979/P.
In hydraulics, the fluid pressure, P in pounds per square inch, is related directly with the force, f in
f
pounds, and inversely with the area, A in square inches. The formula is P = . Assume the force is
A
kept constant.
a.
If the fluid pressure is 5 pounds per square inch when the area is 20 square inches, what is the
fluid pressure when the area is 40 square inches?
The constant force is f = 5(20) = 100. The fluid pressure is 2.5 pounds per square inch when the
area is 40 square inches, P = 100/40.
b.
Write an inverse variation equation for the fluid pressure.
The inverse variation equation for the fluid pressure is P = 100/A.
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Non–Linear Functions
Lesson 5: Inverse Variation
Page 101
SJ Page 47
7.
8.
The weight of a body varies inversely as the square of its distance from
the center of the Earth.
a.
If the radius of the Earth is 4000 miles, how much would a 200pound man weigh 1000 miles above the surface of the earth?
The constant of proportionality k = 200(4000)2 = 3,200,000,000. The
man weighing 200 pounds on Earth would weigh 128 pounds 1,000
miles above the surface of the Earth.
b.
Write an inverse variation equation for the weight of a body.
The inverse variation equation for the weight of a body is W =
3200000000/d2.
Use the graph to the
right, to write an inverse
variation equation.
The inverse variation
equation is y = 500/x.
y
100
50
x
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Practice Exercises
For Exercises 1 through 3, use the given information to solve for the constant of proportionality k and then for
the unknown value. Write an inverse variation equation for each exercise.
1.
If s varies inversely with t and s = 30 when t = 30, what is the value of s when t = 10?
The value of k is 900. Therefore, y = 90 when x = 10. The inverse variation equation is s = 900/t.
2.
If y varies inversely with the fourth power of x and y = 2 when x = 3, what is the value of y when x = 0.1?
The value of k is 162. Therefore, y = 1,620,000 when x = 0.1. The inverse variation equation is y = 162/x4.
3.
If j varies inversely with the square of l and j = 16 when l = 4, what is the value of j when l = 8?
The value of k is 256. Therefore, j = 4 when l = 8. The inverse variation equation is j = 256/l2.
4.
The current, I in amps, produced by a battery varies inversely to the resistance, R in ohms, of the circuit
to which the battery is connected.
5.
a.
If the current is 0.25 amps when the resistance is 10,000 ohms, what will the current be if the
resistance is reduced to 2500 ohms?
The constant of proportionality is k = 10,000(0.25) = 2500. The current when the resistance is
reduced to 2500 ohms is 1 amp, I = 2500/2500.
b.
Write an inverse variation equation for the current of the battery.
The inverse variation equation for the current of the battery is I = 2500/R.
The intensity, I, of light observed from a source of constant luminosity varies inversely as the square of
the distance, d, from the object.
a.
If the intensity of a light is 0.1499 lumens when the light source is 1.1 meters away, what is the
intensity of the light if the source is 3 meters away? Round all answers to four decimal places.
The constant of proportionality is k = 0.1499(1.1)2 ≈ 0.1814. The intensity of the light when the
source is 3 meters away is 0.0202 lumens, I = 0.1814/32.
b.
Write an inverse variation equation for the intensity of light, I, a distance d from the source.
The inverse variation equation for the intensity of the light is I = 0.1814/d2.
Non–Linear Functions
Lesson 5: Inverse Variation
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6.
Lengths of radio waves vary inversely with radio wave's frequency.
a.
Radio station WJHU broadcasts their FM signal with a frequency of 88.1 MHz and has a
wavelength of approximately 3.4 meters. Boston's famous WRKO AM radio station broadcasts
their signals with a frequency of 0.680 MHz. What is the wavelength of WRKO's broadcasts?
NOTE: Round your k value to the nearest whole number and the wavelength to the nearest tenth
of a meter.
The constant of proportionality is k = 3.4(88.1) ≈ 300. The wavelength of WRKO's broadcasts are
441.2 meters, w = 300/0.680.
b.
Write an inverse variation equation for the wavelength of radio waves.
The inverse variation equation for the length of radio waves is w = 300/F.
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Outcome Sentences
Inverse variation is
I know when a problem is about inverse variation because
For inverse variation, y ___________________________________________________________as x
The opposite of inverse variation is
I still need help with
Non–Linear Functions
Lesson 5: Inverse Variation
AIIF
Page 105
Teacher Reference
Lesson 5 Quiz Answers
1.
The value of k is 60. Therefore, a = 600 when b = 0.1. The inverse variation equation is a = 60/b.
2.
The value of k is 15. Therefore, y = 1.5 when x = 100. The inverse variation equation is y =
3.
It will take 7 workers 3 hours to unload the same cargo jet.
15
.
x
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Page 106
Lesson 5 Quiz
Name:
For Problems 1 and 2, use the given information to solve for the constant of proportionality k and then for the
unknown value. Write an inverse variation equation for each problem.
1.
If a varies inversely with b and a = 15 when b = 4, what is the value of a when b = 0.1?
2.
If y varies inversely with the square root of x and y = 3 when x = 25, what is the value of y when x = 100?
3.
The amount of time it takes to unload a cargo jet varies inversely with the number of workers
unloading the jet. If 3 workers take 7 hours to unload the cargo jet, how long will it take 7 workers to
unload the same cargo jet?
Non-Linear Functions
Lesson 6: Exponential Functions
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Page 107
Lesson 6: Exponential Functions
Objectives
• Students will be able to determine the difference between power functions and exponential functions
• Students will be able to identity exponential functions involving growth and decay
• Students will be able to write functions involving exponential applications
• Students will be able to graph exponential functions
• Students will be able to model exponential functions using exponential regression
• Students will be able to solve exponential functions involving applications
Essential Questions
• How do exponential functions behave in real–world applications?
Tools
• Student Journal
• Dry-erase boards, markers, erasers
• Graphing calculator and view screen
• Poster paper
• Construction paper
Warm Up
• Problems of the Day
Number of Days
• 3 days
Vocabulary
Base
Growth
Exponent
Decay
Exponential function
Exponential regression
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Page 108
Teacher Reference
Setting the Stage
Use your favorite grouping strategy to place the class in groups of 4. Place the Setting the Stage transparency
on the overhead projector. Tell the class that the diagram under the first table could represent something such
as arranging oranges in a square while the diagram under the second table could represent a population tree
where each generation doubles. Tell the class to discuss and answer the questions from the transparency. Give
each group a piece of poster paper. Tell the class to answer their questions on the poster paper. Give the class 5
to 7 minutes to answer the questions and place their results onto the poster paper. After the time has expired,
have each group share their results and findings with the class. Some answers to the questions will vary. Here
are some sample responses.
1. The function rule for the left table is y = x 2 while the function rule for the right table is y = 2 x .
2. The similarities are that each rule has a base, an exponent, the number 2, and the variable x.
3. The first function rule would be classified as a quadratic and the second would be classified as an
exponential. Students may not give this response but may correctly guess it since the lesson is about
exponential functions.
4. The conclusion about the function rule for the second table is that the values of y double because the
base is 2.
Discuss with the class the difference between the various group rules and how they would classify the two
rules (functions) they wrote. Questions such as, "Are the functions from the same classification?" and "How
would you classify each function rule you wrote?" The class should realize that the first function rule can be
classified as a power function as well as a quadratic function, while the second function rule is classified as an
exponential function. The students may not know the classification of the second function unless they make a
connection that it might have something to do with the name of the lesson.
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Non-Linear Functions
Lesson 6: Exponential Functions
Page 109
Setting the Stage
x
0
1
2
3
4
5
6
7
8
Transparency
y
0
1
4
9
16
25
36
49
64
x
0
1
2
3
4
5
6
7
8
y
1
2
4
8
16
32
64
128
256
1.
Write a function rule for each table above.
2.
Discuss, in your group, the similarities and the
differences between each function rule.
3.
How would you classify each function rule (linear,
quadratic, power, etc)?
4.
What are your conclusions about the function rule for
the second table?
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Page 110
Teacher Reference
Pre-Reading
Complete the following pre-reading with the students for the first part of the activity on growth.
• Write the word “Bacteria” on the center of the board or overhead. Circle the word and then ask the class to
list other words or ideas that relate to the word. As the students give words, place the words around the
outside of the circle. You are creating a word web and students should be familiar with this from English.
Spend about five minutes discussing the word.
• Write the words "respiratory", “bacteria”, and “lungs” on the over head and ask students why the words
“respiratory”, “lungs”, and “bacteria” may be in the same paragraph. Sample responses might be: “The
lungs are part of the respiratory system; bacteria can get into the lungs and cause serious problems to the
respiratory system.”
• Explain to students that they will be modeling the effect of bacteria on the respiratory system (lungs) and
that they will be modeling this effect with mathematics.
• Remind students of the three methods of representing relationships: numeric, graphic, algebraic. Have
volunteers describe each method.
Activity 1
For the first part of this activity, students will model an exponential growth function. You could let the class
know that different bacteria grow at different rates and for this activity, our rates are for modeling purposes
only and may not reflect actual growth rates. There are 6 steps involved with the "experiment":
• Experiment Step 1: Working with a partner, have the students cut out 64 red squares of construction
paper. Another color may be used. The squares should be 1 square inch or 1 square centimeter in size.
• Experiment Step 2: Have the class cut out the lungs template at the end of the activity.
• Experiment Step 3: Have students place one red square on the lungs so that it is contained within the
lungs. This represents the initial amount of bacteria, a single cell. Tell the class that each hour the bacteria
doubles.
• Experiment Step 4: Walk around to each pair to see their progress. You may want to ask the pairs, “How
long do you think it will take until all of your squares have been placed down on your lungs?”
• Experiment Step 5: Check to make sure the pairs are recording the information correctly. Each pair should
have similar results.
• Experiment Step 6: Ask the students if they see any similarities with their results and either one of the
tables from the Setting the Stage. The students should realize that their table matches the second table from
the Setting the Stage transparency.
Have the pairs answer Exercises 1 through 5 and ask for students to share their results with the class. Ask the
class, "How important is it to obtain medical assistance if problems persist in your respiratory system?" The
class should realize that since the bacteria are doubling every hour that if left unchecked could cause major
health issues.
After the class completes the bacteria portion of the activity, ask them, "If there is bacteria growing and
duplicating in a respiratory system, how would one remove them before the person gets real sick?" This is a
good time to do the pre–reading with the class.
• Write the word “Antibiotics” on the center of the board or overhead. Circle the word and then ask the class
to list other words or ideas that relate to the word. As the students give words, place them around the
outside of the circle forming a word web. Spend about five minutes discussing the word.
Non-Linear Functions
Lesson 6: Exponential Functions
AIIF
Page 111
•
•
Write the word “kidney” on the overhead and ask students why the words “kidney” and “antibiotic” may
appear in the same paragraph. You may want to discuss the function of the kidneys with your students.
You could share information such as, "The kidneys are used to filter waste products and extra water from
the blood. The kidneys filter about two quarts of waste per day." For antibiotic you could say, "Antibiotics
kill bacteria to help prevent people from becoming sicker or to eliminate the disease completely."
Explain to students that in the next portion of the activity they will model the effect of the kidney on
antibiotics in the bloodstream with mathematics.
For the second portion of this activity, students will model an exponential decay function. There are 4 steps
involved with the "experiment":
• Experiment Step 1: Have the students work with a partner. Students should cut out 40 red and 20 blue 1
inch squares of construction paper. (Note: Students can use the red squares from the first portion of the
activity and another color.) The students will need to remember which color represents the blood and
which color represent the antibiotics.
• Experiment Step 2: Have the students place 20 red squares and 20 blue squares into a container (paper bag
or box.) This represents a bloodstream that is half blood and half antibiotics. Although in real life the
bloodstream would not contain 50% antibiotics, this will produce a model quickly that represents the way
antibiotics leave the bloodstream. Walk around to each pair to see their progress. You may want to ask the
pairs, “In reality a person could not have 50% of the blood stream filled with antibiotics, but why might
we model 50%?”
• Experiment Step 3: Shake the container and randomly remove 10 squares. Replace them with 10 red
squares. Check to make sure the pairs are recording the information correctly. Each pair should have
similar results.
• Experiment Step 4: The students should repeat Step 3 ten times. The amount of antibiotics in the blood
stream should be decreasing. Some groups might have zero antibiotics left in the blood stream.
Before the students complete the exercises you may want to remind them that we use exponents to show
repeated multiplication. For example, 34 means (3)(3)(3)(3). Ask the class, "What does 3x mean?" Have the
students work in pairs on Exercises 6 through 9. Have volunteers share their results with the class.
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Activity 1
Bacterial Growth Respiratory System Model
Respiratory sicknesses (infections), such as bronchitis and pneumonia, are
caused by bacteria. Once bacteria gets in our lungs, they can duplicate
(reproduce) at a certain rate. The following experiment will model the amount
of bacteria present over time.
In this experimental model, we will use small construction paper squares of one
color to represent the bacteria.
Experiment Step 1:
Cut out 64 red construction paper squares. Each square
should be the same size and shape. The best size is 1
inch by 1 inch or 1 centimeter by 1 centimeter. Use a
ruler to draw the squares before cutting.
Experiment Step 2:
Cut out the lungs template at the end of the activity.
Experiment Step 3:
Place one red square on the lung template (any where inside the lung area.) This
represents the initial amount of bacteria, a single cell. Note: Bacteria are actually very
small in size. A single cell of bacteria is about 1/10,000th of a centimeter.
Experiment Step 4:
Every minute, add enough red squares to double the amount you had previously. This
represents the bacteria duplicating (reproducing itself) every hour. While you are
waiting for each minute to end, count out the necessary squares that you will be adding
for the next minute. Also, record the time and amount of bacteria present in the lungs in
the table provided below.
Experiment Step 5:
Repeat Step 4 until all 64 squares have been placed "in" your lungs.
Experiment Step 6:
You should realize that your table matches the table from the Setting the Stage
transparency.
Table 1: Bacterial Growth Experiment
Hour
0
1
2
3
4
5
6
Bacteria Count
1
2
4
8
16
32
64
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Non-Linear Functions
Lesson 6: Exponential Functions
Page 113
SJ Page 52
1.
Create a scatter plot of the hours
compared to the number of bacteria in the
lungs. What type of pattern occurred in the
scatter plot graph?
Bacteria Growth Experiment
24
22
Answers will vary. In general, the students
should describe the points following an
increasing pattern from left to right. It
should be a non–linear pattern. The
students may describe this as a curve.
What is the rate that the bacteria are
growing?
The students should be able to determine
that the bacteria's rate is doubling each
hour.
3.
Graph a scatter plot of your data on a
graphing calculator. Set the window range
to an x–minimum of –2, x–maximum of 7, x–
scale of 1, y–minimum of –2, y–maximum of
100, and y–scale of 10. Is the scatter plot
linear? If not describe the shape of the
graph.
18
16
Bacteria Count
2.
20
14
12
10
8
6
4
2
0
0
1
2
3
4
5 6
Hour
7
8
9
10
Answers may vary. A sample answer might be: "No, the scatter plot is not linear. The scatter plot
looks like half a parabola. Walk around to help pairs as needed when students are completing the table
and graphing the data.
4.
How many bacteria do you expect to be in the lungs after a 24 hour period? How might you calculate
this value?
Sample response: To calculate the value I would use an exponential function with 2 as the base and 24
as the exponent. There should 224 or 16,777,216 bacteria in the lungs after a 24-hour period.
5.
Approximately how many hours will it take until there are 1 trillion (1,000,000,000,000 or 1 x 1012)
bacteria in the lungs? NOTE: The graphing calculator may display 1 trillion as 1.0 E12.
It will take approximately 40 hours until 1 trillion bacteria are in the lungs.
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Antibiotic Decay in the Blood Stream Experimental Model
To help cure illnesses antibiotics and/or medicines taken into
the body are circulated throughout the body by the
bloodstream. The kidneys take the drugs out of the blood. We
saw, from the first part of the activity, how bacteria can
duplicate and create enormous amounts of themselves in a
relative short period of time. Bacteria left unchecked can cause
major health problems. Sometimes the only way to become
healthy again is by the use of antibiotics. The following
experiment will model the amount of antibiotics left in the
bloodstream over time.
In this experimental model, we will use small construction paper squares of one color to represent the blood
and small construction paper squares of another color to represent the antibiotics.
Experiment Step 1:
Cut out 40 red construction paper squares and 20 blue construction paper squares.
Each square should be the same size and shape. The best size is 1 inch by 1 inch. Use
a ruler to draw the squares before cutting.
Place 20 red squares and 20 blue squares in a container (bag or box). This represents
a bloodstream that is half blood and half antibiotics. Although in real life the blood
stream would not consist of 50% antibiotics, this will produce a model quickly that
represents the way drugs leave the bloodstream.
Shake the container and randomly remove 10 squares. Replace them with 10 red
squares. Determine how many blood squares and antibiotic squares are now in the
container. Place this information in Table 1 below. This step models the kidneys
randomly cleaning one quarter of the blood each hour.
Repeat Step 3 ten times. Place the information for each cleaning cycle in Table 2,
Antibiotics Decay Experiment, below.
Experiment Step 2:
Experiment Step 3:
Experiment Step 4:
Table 2: Antibiotics Decay Experiment
Hour
0
1
2
3
4
5
6
7
8
9
10
Blood Count
20
Antibiotic Count
20
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Non-Linear Functions
Lesson 6: Exponential Functions
Page 115
SJ Page 54
Answers will vary. In general, the students
should describe the points following a
decreasing pattern from left to right. It
should be a non–linear pattern. The students
may describe this as a curve.
7.
Create a transparency copy of your graph.
Place all the transparencies from each group
on the overhead at one time and line up the
axes. What do you notice about the graph?
Have each pair copy their graph onto a
transparency and then align these on top of
each other on the overhead so that they can
see that most of the groups found a similar
pattern. This should confirm to the students
that it is a non–linear pattern decreasing
from left to right eventually reaching zero.
8.
Antibiotic Decay Experiment
Create a scatter plot of the hours compared
to the number of antibiotics left in the
bloodstream. What type of pattern occurred
in the scatter plot graph?
24
22
20
18
Antibiotic Count
6.
16
14
12
10
8
6
4
2
0
0
1
2
3
4
5 6
Hour
7
8
9
10
If no new antibiotics are added, what would the graph do if we continued with the experiment?
The students should be able to describe that the data or pattern will eventually reach zero and stay
there.
9.
Graph a scatter plot of your data on a graphing calculator. Set the window range to an x–minimum of
–2, a x–maximum of 12, a y–minimum of –2, and a y–maximum of 24. Is the scatter plot linear? If not
describe the shape of the graph.
Walk around and help pairs as needed to complete the table and graph of the data. There may be some
groups of students who have outliers that do not match the data. You may want to discuss with the
class why that may happen.
10.
Graph y = 20(0.75)x on the same graph as the scatter plot. Describe how the graph of y = 20(0.75)x fits
the data from the scatter plot. Answers will vary. This function should model the data fairly well.
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Cut Here
SJ Page 55
Lungs Template
Non-Linear Functions
Lesson 6: Exponential Functions
AIIF
Page 117
Teacher Reference
Activity 2
In this activity, students will investigate the graphs and terminology of exponential functions. Students should
be familiar with the terms base and exponent. Determine students understanding of these terms by placing
the function rules, from the Setting the Stage, on the board or on a blank transparency on the overhead. The
class should be able to tell you for the first function rule that x represents the base and 2 is the exponent, y = x 2 .
For the second function rule, the class should be able to tell you that x represents the exponent and 2 is the
base, y = 2 x . The class could also develop the standard form of an exponential function from the bacteria
growth from the last activity.
Hopefully, during your discussion of the differences between the function rules in the Setting the Stage you
touched upon the topic that the exponent, for the function rule for the second table, was a variable not a
constant. Tell the class that when the exponent contains a variable we have an exponential function. Have the
class investigate the graphs of exponential functions in their groups. You may want to use the examples below
to start. It is best if the students study one form, growth, and then move to the other form, decay. They did see
an example of each type from the last activity.
Sample exponential functions related to growth
•
y = 2 x and y = 5x
•
y = 2 x and y = 3(2 x )
•
y = 2 x and y = 2 x + 3
•
y = 2 x and y = 2 x − 3
•
y = 2 x and y = 3(2 x ) + 1
Have the students form groups of three or four. Pose the following questions and have the students report
their findings. Allow about 2 minutes a question for discussion. Some guiding questions for their groups are:
"What do you notice about the graphs as the base increases?" and "What do you notice about the graphs when
the function is multiplied by a constant or a constant is added or subtracted?" Explain to the students that the
graphs could have a vertical translation caused by the multiplication or addition/subtraction by a constant.
Ask the students, "From our previous lesson about power functions, how did we obtain a horizontal
translation of a function?" The students may remember that they had to add or subtract a value from x in order
to obtain a horizontal translation. The same is true for exponential functions. These groups of exponential
functions represent growth because y increases "exponentially" as x increases.
Ask the groups, "How might we write an exponential function which decreases or decays?" Have students
share their responses with the class. You might want to have a volunteer write the responses on the board. The
students can use their graphing calculator to determine which methods produce a decreasing exponential
function. Note: The functions should be exponential in nature, meaning the exponent should be a variable.
Now have the class, in their groups, investigate the sample exponential functions below. These functions
represent decay.
Sample exponential functions related to decay
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Page 118
x
•
⎛1⎞
y = 2 − x and y = ⎜ ⎟
⎝3⎠
x
•
⎛1⎞
y = 3− x and y = ⎜ ⎟
⎝2⎠
⎛3⎞
y = 4 − x and y = ⎜ ⎟
⎝4⎠
x
•
Pose the following guiding questions for the groups to answer, "What do you notice about the graphs of the
exponentials with negative exponents compared to the exponentials with fractions as bases?" and "Are
exponentials with negative exponents and exponentials with fractions both examples of decay? Explain."
Sample responses might be:
• The graphs of the exponentials with negative exponents are similar to the graphs of exponentials with
fractions as bases.
• Yes, exponentials with negative exponents and exponentials with fraction bases are both examples of
decay since as x increases, y decreases exponentially. Note: Since the graphing calculator will give a
base with a fraction for exponential regression, the lessons will focus on exponentials with a fraction for
the base instead of exponentials with negative exponents.
Write the standard form of an exponential function, y = Cabx , on the board. Note: y represents the function or
dependent variable, C is a constant and also known as the initial amount when x = 0, a is the numeric base, b is
a constant, and x is the variable exponent as the independent variable. Place the Activity 2 transparency on the
overhead projector. Ask for student volunteers to label the graphs as growth, decay, growth with initial amount,
or decay with initial amount. The idea here is that students can look at an exponential function and should be
able to determine if the function represents growth or decay and if there was an initial amount or not. Ask the
class, "How many intercepts are there for exponential functions? What is the standard format for the y–
intercept of an exponential function?" The key goal here is that the class realizes there is only one intercept, the
y–intercept, and its standard format is (0, C). But in general, we use the standard form of y = Cabx to define an
exponential function.
Use the sample examples below and model, or have volunteers model, how to determine the y–intercept.
Review how the students found the y–intercept when they graphed equations of the form y = mx + b. Students
should remember that for the y–intercept the x coordinate has a value of zero. Also, have the volunteer state
the type of graph the function represents.
Sample Exponential Functions
y = 4 x ; y–intercept of (0, 1), type: growth
⎛1⎞
y = 4⎜ ⎟
⎝3⎠
2x
; y–intercept of (0, 4), type: decay with initial amount
y = 100(4 3 x ) ; y–intercept of (0, 100), type: growth with initial amount
y = 12(5x + 1 ) ; y–intercept of (0, 60), type: growth with initial amount
Make sure the Activity 2 transparency is displayed while to class works on the exercises as a reference. Have
the class work in pairs on Exercises 1 through 6. Divide the board into 6 sections and have 6 students share
their results with the class on the board. Ask the volunteers some guiding questions such as, "How did you
Non-Linear Functions
Lesson 6: Exponential Functions
AIIF
Page 119
find the y–intercept for your function?" and "How did you determine the type of graph and use it to draw your
rough sketch of the function?" Some of the students might have answers such as, "I found the y–intercept by
setting x = 0 and then evaluating the function." and "I entered the function into my graphing calculator and
viewed that the graph was increasing or decreasing so I knew it was either a growth or a decay type of graph."
AIIF
Page 120
Activity 2
Transparency
A. y = Cax, a > 1, C = 1
B. y = Cax, a > 1, C > 1
C. y = Ca–x, a > 1, C > 0
D. y = Cax, 0 < a < 1, C > 0
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Non-Linear Functions
Lesson 6: Exponential Functions
Page 121
Activity 2
SJ Page 57
In this activity, you will determine the y–intercept, determine the type of graph, and draw a rough sketch of
exponential functions.
For Exercises 1 through 4:
a. Determine the coordinates of the y–intercept
b. Type of graph: growth or decay
c. Draw a rough sketch of the exponential function on the grid provided. Note: set grid scale
appropriately.
1.
y
y = 12 x
2300
2200
2100
2000
1900
a.
b.
1800
The coordinates of the y–intercept are (0, 1).
The graph is a growth type.
1700
1600
1500
1400
1300
1200
1100
1000
900
2.
⎛1⎞
y =⎜ ⎟
⎝8⎠
x
a.
b.
The coordinates of the y–intercept are (0, 1).
The graph is a decay type of graph.
800
700
600
500
400
300
200
100
-5
-4
-3
-2
x
-1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
y
2300
2200
2100
3.
⎛1⎞
y = 9⎜ ⎟
⎝5⎠
2x
2000
1900
1800
1700
1600
1500
1400
1300
a.
b.
The coordinates of the y–intercept are (0, 9).
The graph type is decay with initial amount.
1200
1100
1000
900
800
700
600
4.
x
500
y = 7(4 )
400
300
200
100
a.
b.
The coordinates of the y–intercept are (0, 7).
The graph type is growth with initial amount.
-5
-4
-3
-2
y
y
2300
2300
2200
2200
2100
2100
2000
2000
1900
1900
1800
1800
1700
1700
1600
1600
1500
1500
1400
1400
1300
1300
1200
1200
1100
1100
1000
1000
900
900
800
800
700
700
600
600
500
500
400
400
300
300
200
200
100
-5
-4
-3
-2
-1
x
-1
100
x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
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SJ Page 58
For Exercises 5 through 7, state the y–intercept and the type of graph.
5.
6.
y
y
2300
2300
2200
2200
2100
2100
2000
2000
1900
1900
1800
1800
1700
1700
1600
1600
1500
1500
1400
1400
1300
1300
1200
1200
1100
1100
1000
1000
900
900
800
800
700
700
600
600
500
500
400
400
300
300
200
200
100
-5
-4
-3
a.
-2
-1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
-5
7.
-4
-3
-2
a.
The coordinates of the y–intercept are
(0, 6).
The graph type is exponential decay
with initial amount.
b.
100
x
b.
y
2300
2200
2100
2000
1900
1800
1700
1600
1500
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
200
100
-5
a.
b.
-4
-3
-2
-1
x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
The coordinates of the y–intercept are (0, 3.5).
The graph type is exponential decay with initial amount.
-1
x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
The coordinates of the y–intercept are
(0, 1).
The graph type is exponential growth.
Non-Linear Functions
Lesson 6: Exponential Functions
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Page 123
Teacher Reference
Activity 3
In this activity, students will investigate real–world applications of exponential functions. Explain to the class
that in the last activity, they looked at the functions of the form y = Cabx, where C was the initial value, a > 0.
The values of b determined if the function was for growth (b > 0) or decay (b < 0).
Either you or a volunteer model the following sample problem.
The future value, S, of an investment that is compounded monthly; can be determined by the function
S = P(1 + i )n , where P is the amount invested, i is the interest rate per month (rate/12) as a decimal value,
and n represents the number of months the money has been invested. Determine the future value of a
$15,000 investment, invested at the rate of 3%, for 60 months.
From the information given, we see that P = $15,000, i = rate/12 = 0.0025, and n = 60. Substituting in these
values and rounding to the nearest cent we get S = 15000(1.0025)60 = $17,424.25.
Have a second student volunteer model the following problem.
From the bacterial growth problem from the first activity, we saw that the bacteria doubled every hour
and the number of bacteria after t hours was given by the function y = 2 t . If a different strand of bacteria
was present such that it quadrupled every hour, the function would be y = C (4t ) , where C the initial
amount of bacteria. Calculate the number of bacteria present after 24 hours if there are initially 10
bacteria present.
From the information given, we see that C = 10, t = 24. Substituting in these values we get that the number of
bacteria present after 24 hours is y = 10(4)24 ≈ 2.81475 X 1015. There is almost 3 quadrillion bacteria. Ask the
class how this amount compares to their result in Exercise 4 from the first activity. Ask the class to calculate the
ratio of the amount of these bacteria to the amount in Exercise 4 Activity 1. The class should get that there is
167,772,160 times more bacteria from this problem compared to Exercise 4 Activity 1. These are mind boggling
numbers.
Divide the class into teams of two. This might be different partnerships than before. Have the pairs work on
Exercises 1 through 6. As the students finish their problems have pairs check their results with one or more
other pairs. Bring the class together and have volunteers share their results on the board or overhead projector
on any problems that the class had trouble. While the students are working, circulate to provide help, ask
guiding questions, and provide encouragement. Remind the students that when using their calculators they
should always double check their results.
Note: For the Practice Exercises, you may want the class to work in pairs for Practice Exercise 4 through 8. You
may want to make Exercise 7 optional or a bonus exercise.
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Page 124
SJ Page 59
Activity 3
In this activity, you will solve real–world exponential problems.
1.
Your grandparents put $10,000 in an investment
account, which collects interest four times a year,
when you were born for your college education. The
future value of your college education fund can be
y
2300
2200
2100
2000
1900
1800
1700
1600
1500
4t
determined by the function S = 10000(1.0375) , where
t represents the number of years for the investment.
How much money will you have available when you
start college? Assume you will be 18 years old when
you start college. Draw a rough sketch of the
investment; set axis scales accordingly.
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
200
100
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
I will have $141,626.20 available when I start college. Note: y–axis scale is 6,000:1
2.
Viruses can produce many more offspring than bacteria per infection. Some viruses produce at an
t
exponential rate related to the function v = C (100) h , where v represents the number of viruses, C
represents initial population of viruses, t represents amount of time in hours, and h is the number of
hours to produce a new generation. How many viruses will be present after 24 hours if there initially
were 5 viruses and the viruses produce a new generation every 4 hours?
There will be 5,000,000,000,000 viruses after 24 hours.
3.
It has been determined that a certain city has been growing exponentially over the last 20 years
according to the function P = P0 (1 + r )t , where P represents the town's population, P0 is the initial
population, r is the rate at which the town's population is increasing, and t is the amount of time in
years that the town has been increasing. If the town initially had 450 people 20 years ago and they now
have 1,443 people, what was the rate of increase in population over the last 20 years? Round your
answer to the nearest whole percent.
The rate of increase for the population over the last 20 years was approximately 6%.
4.
A local retail store has determined that its sales could grow exponentially based on the amount they
spend on advertising each week by the function s = C (1.15)w , where s represents the number of sales
per week, C represents their initial sales before advertising began, w represents the number of
consecutive weeks they advertised. If the store averaged 125 sales per week before advertising began,
how many sales can they expect to have, each week, after advertising for 4 consecutive weeks? Round
your answer down to the nearest whole sale.
The store can expect to have 218 sales each week after advertising for 4 weeks.
AIIF
Non-Linear Functions
Lesson 6: Exponential Functions
Page 125
5.
– ( 0.693t /T )
SJ Page 60
The radio active decay of a material is given by the function A = A0 e
, where A0 is the initial
amount of the material, t is the amount of time in years, and T is the half–life of the radio active
material. Plutonium 240 has a half life of 6540 years. If a nuclear power plant started with 100 pounds
of Plutonium 240, how much would be left after 20 years? How many ounces of plutonium decayed
during the 20 years? Round your answers to the nearest hundredth pound and ounce.
There would be 99.79 pounds still left after 20 years. There was approximately 3.36 ounces of
plutonium that decayed during the 20 years.
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Page 126
Teacher Reference
Activity 4 (optional)
In this activity, students will learn and use exponential regression to determine the exponential function
represented by a set of data. Review with students, if necessary, how to enter data in their graphing calculator.
Since the students have used linear and quadratic regression models, they should be familiar with all aspects
of regression except for maybe how to obtain the results for exponential regression.
Have a volunteer model exponential regression by using the following data and the graphing calculator view
screen.
Time in
minutes (t)
Temperature
°C (T)
0
5
10
15
20
25
30
35
40
98.4
82.9
73.4
66.3
60.9
56.0
52.3
49.0
46.7
After the class has entered the data, the class should press the following key sequence to obtain the
exponential regression function STAT ~ Ê Í. The following screen shots coincide with all but the first
key pressed.
You could have the students complete a scatter plot of the data and have them store the exponential regression
in the Y= editor by using the following key sequence: STAT ~ Ê  ~ Í Í Í. The following
screen shots show the majority of the above key sequence along with the graph and scatter plot.
Have the class continue to work in their pairs for Exercises 1 through 3. Have students share their results using
the graphing calculator view screen.
AIIF
Non-Linear Functions
Lesson 6: Exponential Functions
Page 127
Activity 4
SJ Page 61
In this activity, you will use exponential regression to obtain an exponential function from real–world data.
1.
The following data table represents the daily costs of commuting (driving to work) versus the amount
of commuters (people who drive to work) for a large metropolitan area.
Cost (in $)
Commuters
10
225,000
15
145,000
20
110,000
25
68,000
30
35,000
35
13,000
40
8,000
45
5,600
50
2,500
What type of graph does the data model?
a.
The data models a decay type of graph.
b.
What is the exponential regression function? Round values to three decimal places.
x
The exponential regression function is y = 911282.749 ( 0.891) .
How many commuters would you expect if they had to pay $75.00 each day in commuting
expenses? Round your answer to the nearest commuter.
c.
There would be about 158 commuters.
2.
The following data table represents the population of the United States from the years 1790 through
2000, where year 0 = 1790, 1 = 1820, etc.
Year
Population
(in
millions)
a.
0 (1790)
3.93
1 (1820)
9.64
2 (1850)
23.19
3 (1880)
50.16
4 (1910)
91.97
5 (1940)
131.67
6 (1970)
204.05
7 (2000)
281.42
What type of graph does the data model?
The data models a growth type of graph.
b.
What is the exponential regression function? Round values to four decimal places.
x
The exponential regression function is y = 5.7988 ( 1.8347 ) .
c.
Using this exponential equation, what might you predict will be the size of the U. S. population
in the year 2060? Round your answer to the nearest ten thousandths. Note: Remember our
current units for population is in millions. Sample response: There would be about 1,365,730,000
people or 1,365.73 million people.
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Page 128
SJ Page 62
3.
The following table represents the early production of crude petroleum in the United States.
Year
Oil
Production
(in barrels)
a.
0 (1859) 10 (1869) 20 (1879)
30 (1889)
40 (1899)
2,000
4,215,000 19,914,146 35,163,513 57,084,428
What type of graph does the data model?
The data models a growth type of graph.
b.
What is the exponential regression function? Round values to three decimal places.
x
The exponential regression function is y = 34599.182 ( 1.254 ) .
c.
U. S. oil production peaked in 1970. What could you predict was our country's peak output of oil
in 1970? Round your answer to the nearest whole barrel.
The peak output of oil production in the U. S. was approximately 2,819,000,000,000,000 or
2.819X10 15 barrels of oil.
d.
The actual U. S. oil production in 1970 was approximately 3,500,000,000 barrels. What can you
say about your predicted value of production compared to the actual value of production?
Answers will vary. A sample response might be: The values differ by quite a lot.
e.
What suggestion would you make on limiting the use of your exponential regression function?
Answers will vary. A sample response might be: I think the function should be limited to within
only a few years from the last date of data collection.
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Non-Linear Functions
Lesson 6: Exponential Functions
Page 129
Practice Exercises
SJ Page 63
For Exercises 1 and 3:
a.
Determine the coordinates of the y–intercept.
b.
Type of graph: growth or decay.
c.
Draw a rough sketch of the exponential function on the grid provided. Note: set grid scale
appropriately.
1.
( )
y = 5 3x .
a.
b.
y
2300
2200
The coordinates of the y–intercept are (0, 5).
The graph type is growth. Note: y−scale is 5:1
2100
2000
1900
1800
1700
1600
1500
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
200
100
-5
-4
-3
-2
x
2.
x
-1
⎛1⎞
y = 9⎜ ⎟ .
⎝6⎠
a.
The coordinates of the y–intercept are (0, 9).
b.
The graph type is decay with initial amount.
Note: y−scale is 100:1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
y
2300
2200
2100
2000
1900
1800
1700
1600
1500
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
200
100
-5
-4
-3
-2
x
-1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
y
2300
3.
2200
( )
2100
y = 3 42 x .
a.
b.
2000
1900
1800
The coordinates of the y–intercept are (0, 3).
The graph type is growth with initial amount.
1700
1600
1500
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
200
100
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
7
8
9
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17
18
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SJ Page 64
4.
On January 15th, 2009, the world's population was 6.75 billion people. It is predicted that it will take just
44 years for the world's population to double. What is the rate, per year, at which the world population is
increasing? Round your answer to the nearest tenth of a percent. Note: Use P = P0 (1 + r )t . Refer back to
Activity 3, Exercise 3.
The world's population is increasing at an approximate rate of 1.6% per year.
5.
A biologist is conducting an experiment testing a new antibiotic on a certain strain of bacteria cells.
According to the biologist's calculation, the cells are dying (decaying) at a rate given by the
function L = ae −0.223t , where L represents the amount of cells left after time t (in minutes) and a represents
the initial amount of bacteria cells present before the antibiotic is applied. How many bacteria cells are
present ten minutes after the antibiotic was applied if there initially were 10 million bacteria cells? Round
your answer to the nearest whole cell. Use 2.178 for the value of e.
There are approximately 1,075,284 cells left 10 minutes after the antibiotic was applied.
6.
A person invests $15,000 into an interest bearing account. After 10 years the person's investment is now
worth approximately $25,966. Determine the annual interest rate if the future value of an investment can
be determined with the function S = P(1+ r/12)12t, where S is the value of the investment after t years, P is
the amount invested, and r is the annual interest rate. Round your answer to the nearest tenth of a percent.
The annual interest rate is 5.5%.
7.
The intensity of earthquakes is measured by using the Richter scale. We can determine how much more
powerful one earthquake is compared to another earthquake, by the ratios of their intensities. The ratios
of the intensities of two earthquakes can be determined by the function I = 10d, where I is the ratio of
intensities and d is the absolute value of the difference of the intensities of the earthquakes as measured
by the Richter scale. It is estimated that the 2004 Indian Ocean earthquake measured 9.2 on the Richter
scale. In comparison, the earthquake that caused Mt. St. Helen's volcano to erupt on May 18th 1980,
measured 5.1 on the Richter scale.
a.
How much more powerful was the 2004 Indian Ocean earthquake compared to the 1980 Mt. St.
Helen's earthquake? Round your answer to the nearest whole number.
The 2004 Indian Ocean earthquake was 12,589 times more powerful than the 1980 Mt. St. Helen's
earthquake. I = 10 (9.2 – 5.1).
b.
What can you conclude about the difference in the intensities of two earthquakes?
Answers will vary. A sample response might be: "The difference of the intensities of two
earthquakes grows exponentially by a factor of 10.
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Non-Linear Functions
Lesson 6: Exponential Functions
Page 131
SJ Page 65
8.
For the following graph:
a.
State the y–intercept.
The y–intercept for the graph is (0, 10).
b.
State the type of graph.
The graph type is decay with initial amount.
y
2300
2200
2100
2000
1900
1800
1700
1600
1500
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
200
100
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
AIIF
Page 132
SJ Page 66
Outcome Sentences
Exponential growth is
I know an exponential problem when
Exponential decay is
When graphing exponential functions
The part about exponential functions I don't understand is
AIIF
Non-Linear Functions
Lesson 6: Exponential Functions
Page 133
Teacher Reference
Lesson 6 Quiz Answers
1.
a.
b.
c.
The coordinates of the y–intercept are (0, 12).
The graph type is growth with initial amount.
Note: vertical scale is 10.
2.
$
Time (years)
0
5
10
15
20
Value of Investment ($)
25,000
31,268.76
39,109.42
48,916.13
61,181.87
60,000
50,000
40,000
30,000
20,000
10,000
5
3.
t (in hours)
0
8
16
24
32
40
48
Size of virus colony
5
125
3125
78125
1,953,125
48,828,125
1,220,703,125
10
15
20
t
AIIF
Page 134
Lesson 6 Quiz
1.
Name:
y = 3(4x+1).
a.
Determine the coordinates of the y–intercept.
b.
Type of graph: growth, decay, growth with initial
amount, or decay with initial amount.
c.
Draw a rough sketch of the exponential function
on the grid provided. Note: set grid scale
appropriately.
Complete one of the two problems below.
4t
2.
r⎞
⎛
The function S = P ⎜ 1 + ⎟ can be used to determine the
4⎠
⎝
future value of an investment that is compounded
quarterly (4 times per year). If $25,000 is invested at a
rate of 4.5%, how much will the investment be after 5
years, 10 years, 15 years and 20 years? Create a table for
the future value of the investment and create a
connected scatter plot of the investment over the 20 year
period.
Time (years)
3.
Value of Investment ($)
t /8
The rate at which a particular virus duplicates is given by the function v = C ( 25 )
, where v represents
the amount of the virus after t hours and C represents the initial size of the virus colony. Create a table
showing the size of the virus colony every 8 hours for a 2 day period if there is initially a population of 5
virus cells.
t (in hours)
0
Size of virus colony
Non–Linear Functions
Lesson 7: Step Functions
AIIF
Page 135
Lesson 7: Step Functions
Objectives
• Students will be able to determine the rise and run of a step function
• Students will be able to write equations involving step functions by using the floor function
• Students will be able to write equations involving step functions by using the ceiling function
• Students will be able to graph step functions using the int() function
• Students will be able to solve step functions involving real–world applications
Essential Questions
• How do step functions apply to real–world applications?
Tools
• Student Journal
• Setting the Stage transparency
• Dry–erase boards, markers, erasers
• Graphing calculator and view screen
Warm Up
• Problems of the Day
Number of Days for Lesson
• 2 Days
Vocabulary
Rise
Greatest integer
Ceiling function
Run
Floor function
Binary number system
Int () function
Smallest integer
Pitch/slope
AIIF
Page 136
Teacher Reference
Setting the Stage
Place the class in pairs. Make sure each student pair has a dry–erase board and two different colored markers.
Place the Setting the Stage transparency on the overhead projector but cover the bottom portion containing the
parts of a stair stringer. Tell the class that stairs in homes and on wooden porches usually are built with
stringers cut from a single board. Ask the class to draw a set of similar stairs on the grid side of their dry–erase
boards. Have the class wait to draw the dotted line until all the parts of the stringer have been labeled.
Uncover the bottom of the transparency containing the parts of the stair stringer. Have volunteers locate the
place where the parts of the stair stringer belong on the transparency while the rest of the class locates the
parts on the stairs they drew on their dry–erase boards. Now have the class draw the pitch line on their
stringer which connects the top of each tread as shown in the transparency.
Lead a guided class discussion on the similarities and differences between the stringer and finding the slope of
a line. Ask guiding questions such as, "What part or parts of the stair stringer are similar to the formula for the
slope of a line?" and "What does the rise and run of the stair stringer tells us about stairs that will be built with
the stringer?" The key concept here is the class understands that the rise and run of the stringer gives the slope
of the stairs that will be built and eventually people will walk up and down on.
Have the class calculate the slope of their stairs. Tell the class that local building codes for their city determines
pitch (slope) of a set of stairs. Normally, the pitch value is 7/11, meaning 7–inch rise and an 11–inch run. Ask
for students to share the slope of their stairs. Does the pitch of their stringer fall within the "building code"
value? Note: Different cities and states could have different building code requirements that stairs must meet.
Research the building codes in your city and state and tell the class what the building code for stairs are and
that the stairs in the Setting the Stage transparency would have to be modified to meet local building codes.
AIIF
Non–Linear Functions
Lesson 7: Step Functions
Page 137
Setting the Stage
Transparency
Stairs Stringer
Stringers in homes and on
porches are usually cut
from a single board.
Parts of a Stair Stringer:
• Rise
• Run
• Stringer Board
• Pitch Line
AIIF
Page 138
Teacher Reference
Activity 1
Have the class continue to work in their pairs. Ask the class, "How do you think a step function looks?" and
"How might the rise and run of a step function be utilized in graphing a step function?" Have the class draw
(graph) their interpretation of a step function on their dry–erase board. Have students show and explain their
interpretation of what the graph of a step function might look like.
In this activity, students will investigate step functions using their graphing calculator and how step functions
apply to the application of salary. The instructions below are based on a TI-83 or 84 Plus™ graphing calculator.
If you use a different type of graphing calculator, consult the owner's manual for graphing an equivalent step
function.
Tell the class that the graphing calculator has only one function that operates as a step function. This function
is called the int() function and it represents the greatest integer. The int(x) is the greatest integer less than or
equal to x. For example, if x = 3.14 then the int(3.14) = 3. Let the class know that mathematically the correct
term to use is floor function when dealing with the greatest integer of a value. Tell the class it is called the
floor function because the integer portion of the number represents the floor, or smallest value that the number
can have. Another way to view it is that the floor is below us, so we want the integer “below” the give value.
This could help students to remember when to “round down”.
To obtain the int() function to the home screen or to the Y= editor, press the following key sequence:  ~
·. The int( function is displayed. The screen shots below represent the key sequence.
Have the class use their graphing calculator to find the values in the sample exercises. Have a student or
students model the first problem in each problem set from the sample exercises using the graphing calculator
view screen while the class parallels with the second problem in each problem set from the sample exercises.
Note: We'll use the mathematical term floor in the sample exercises and when discussing the greatest integer
function.
Sample exercises
• floor of 3.5 and floor of 4.6 (results should be 3 and 4, respectively)
• floor of –6.8 and floor of –8.9 (results should be –7 and –9, respectively)
•
floor of
29.6 and floor of 12.05 (results should be 5 and 3, respectively)
Model, or have a student model, graphing y = int(x) on the graphing calculator view screen while the class
parallels on their graphing calculator. Tell the students to set the graphing window parameters to standard.
Have the students press the r key and trace the values of x and watch as the y values change for
Non–Linear Functions
Lesson 7: Step Functions
AIIF
Page 139
particular x values. Now, have a student model y = 3int(x) while the class parallels with y = 2int(x). Again,
have the class use the r key to determine how multiplying the greatest integer function by a constant
value affects the x and y values. Ask the students, "How were the values of x and y affected by the
multiplication of a coefficient?" The students should be able to tell you that the x–values were unaffected but
the y–values doubled or tripled when the floor function was multiplied by a coefficient of 2 or 3, respectively.
Have other students model the first problem in each sample exercises below while the class parallels with the
second problem from each of the sample exercises. Circulate to provide additional help to students who are
having difficulty by asking guiding questions or offering encouragement. After each example discuss how the
x- and y–values changed from the basic y = int(x) function.
Sample exercises
• y = int(x)/3 and y = int(x)/2
• y = int(x) + 3 and y = int(x) + 2
• y = int(x) – 3 and y = int(x) – 2
• y = int(3x) and y = int(2x)
• y = int(x/3) and y = int(x/2)
After the modeling has been completed, have the class get in groups of four and write a list of their findings
from the sample exercises. Have the class use the term floor function or greatest integer function when
describing their findings. Lead a discussion with the class on their findings and have groups share their list of
findings. Have the students walk around to see how their list compares with other groups. A list of their
findings might include:
• Dividing the floor function by a number divides the y–values by the same amount (as compared to the
original floor function of y = int(x)).
• Adding or subtracting a constant value to the floor function creates a horizontal translation by the
number of units that is added or subtracted.
• Multiplying the x–values by a number in the floor function decreases the x interval for the y–values by
a factor that x was multiplied by. For example, y = int(x) had an x interval of length 1 while y = int(2x)
has an x interval length of 1/2.
• Dividing the x-values by a number in the floor function increases the x interval for the y–values by a
factor that x was divided by. For example, y = int(x) had an x interval of length 1 while y = int(x/2) has
an x interval length of 2.
Show the class how to graph step
functions from the graphing calculator.
For example, to graph , y = floor(x) or , y
= int(x), there would be a closed circle on
the left endpoint of the interval, but an
opened circle on the right endpoint. The
first graph at the right is how the graph
is displayed on the graphing calculator.
The second graph is how it would
actually look if graphing by hand. Model
this closed and open circle for the endpoints for the intervals:
• –3 to –2
• –2 to –1
• –1 to 0
AIIF
Page 140
•
•
•
0 to 1
1 to 2
2 to 3
Each tick on the x–axis represents 2 units and each tick mark on the y–axis represents 2 units for the second
graph above. Students can see how this works by setting the TblStart to –3 and ΔTbl to 0.1 and viewing the
table of values for y = int(x) on the graphing calculator. Model for the students how to write the intervals for x
which define the y–values. The students could write these as inequalities like they did in the Solving OneVariable Equations unit for one variable inequalities. An example would be –3 ≤ x < –2 or [–3, –2).
Working with their partner, have the students complete Exercises 1 through 3. Students can check their work
with another pair. While the students are working, circulate to provide support, clarifications, and
encouragement. Bring the class together and have students share their results on the board or overhead
projector on any problems that the class had trouble with.
AIIF
Non–Linear Functions
Lesson 7: Step Functions
Page 141
Activity 1
SJ Page 67
In this activity, you will investigate the graphs of step functions using your graphing calculator and the int( )
function which represents the greatest integer function. Mathematically, the greatest integer function is called
the floor function. The int(x) is the greatest integer less than or equal to x. If x = 3.14 then, in function notation,
f (3.14) = int(3.14) = 3. Likewise, mathematically f (3.14) = floor (3.14) = 3, or y = floor (3.14) = 3.
1.
Evaluate the floor function using your graphing calculator. Write the function down as it was entered
in the calculator.
a.
y = floor (23.001)
The floor of 23.001 is 23. I entered int(23.001) into the graphing calculator.
b.
y = 3•floor (15.06)
Three times the floor of 15.06 is 45. I entered 3int(15.06) into the graphing calculator.
c.
y = –6•floor (–34.005)
Negative six times the floor of –34.005 is 210. I entered –6int(–34.005) into the graphing
calculator.
d.
y = 5•floor (–5.045) + 3
Five times the floor of –5.045 plus 3 is –27. I entered 5int(–5.045)+3 into the graphing
calculator.
e.
y = ( floor (−13.45) )
2
The square of the floor of –13.45 is 196. I entered int(–13.45)^2 into the graphing calculator.
2.
Your younger sister wants to earn some money. She asks you if you have any chores she can do. Write
a step (floor) function for each scenario below.
a.
You pay your sister $1.00 for each half hour of work.
y = floor (2x)
b.
You pay your sister $1.00 for each fifteen minutes of work.
y = floor (4x)
c.
Your sister wants $2.50 for each hour of work.
y = 2.5 • floor (x)
d.
Your sister wants $4.00 for each hour of work.
y = 4 • floor (x)
e.
Your sister wants $2.50 for each half hour of work.
y = 2.5 • floor (2x)
AIIF
Page 142
SJ Page 68
3.
Create a table of values, which include intervals for x and values for y, and write a step function for
each of the graphs below.
y
a.
10
5
–10
5
–5
10
x
–5
–10
The step function is y = 3 • floor( x ) .
x intervals
[–4, –3) or
–4 ≤ x < –3
[–3, –2) or
–3 ≤ x < –2
[–2, –1) or
–3 ≤ x < –1
[–1, 0) or
–1 ≤ x < 0
[0, 1) or
0≤x<1
[1, 2) or
1≤x<2
[2, 3) or
2≤x<3
[3, 4) or
3≤x<4
y- values
–12
–9
–6
–3
0
3
6
9
y
b.
10
5
–18
–12
6
–6
12
18
–5
–10
The step function is y = 4 • floor( x / 6) .
x
x intervals
[–18, –12) or
–18 ≤ x < –12
[–12, –6) or
–12 ≤ x < –6
[–6, 0) or
–6 ≤ x < 0
[0, 6) or
0≤x<6
[6, 12) or
6 ≤ x < 12
[12, 18) or
12 ≤ x < 18
[18, 24) or
18 ≤ x < 24
y-values
–12
–8
–4
0
4
8
12
AIIF
Non–Linear Functions
Lesson 7: Step Functions
Page 143
Teacher Reference
Activity 2
In this activity, we will investigate another step function, the ceiling function. Ask the class, "What is the
opposite of the floor?" The students should know that the ceiling is the opposite of the floor. Now ask the class,
"If the floor of 3.14 is 3, what do you think the ceiling of 3.14 equals?" Have a student record the class
responses on the board. If nobody came up with 4, tell the class that the ceiling is one more than the floor. The
ceiling of x is defined to be the smallest integer not less than x. Another way to view it is that the ceiling is
above us, so we want the integer “above” the given value. This could help students to remember when to
“round up”.
Now tell the class, "The graphing calculator only has the int() function which we can use as the floor function.
If the ceiling is one more than the floor, for non–integer values, could we use the int() function to behave like a
ceiling function? If so, would there be restrictions as to when we could use it and when we could not use it?"
The students should be able to state that to use the int() function as a ceiling function, all they would have to
do is to add 1 to the int() function to get the correct results, but the int() function can only be applied to non–
integer values of x. In symbolic notation we would have y = int(x) + 1.
Have the class use their graphing calculator to find the values in the sample exercises. Have a volunteer(s)
model the first problem of each sample exercise using the view screen and overhead projector while the rest of
the class does the second problem from each of the sample exercises. Note: We'll use the mathematical term
ceiling in the exercises and when discussing the least integer function.
Sample exercises
• ceiling of 12.51 and ceiling of 9.16 (results should be 13 and 10, respectively)
• ceiling of –7.98 and ceiling of –4.39 (results should be –7 and –4, respectively)
•
•
ceiling of 39.6 and ceiling of 32.05 (results should be 7 and 6, respectively)
ceiling of 8 and the ceiling of 6 (results should be 8 and 6, respectively)
Tell the class to graph a ceiling function on the graphing calculator, other than just the ceiling of x, using the
int() function is complicated and beyond the scope of this class. The students could graph the ceiling of x using
int(x), but multiples or a shorter interval is beyond the scope of this lesson.
Tell the students that just like everything else they do, mathematicians have symbols for both the floor and
ceiling functions.
└ ┘
Floor
┌ ┐
Ceiling
AIIF
Page 144
Have students model writing the previous exercises using the ceiling symbolic notation while the class
parallels with their same exercises.
Sample exercises
• ceiling of 12.51 and ceiling of 9.16 (results should be y = ⎢⎡12.51⎥⎤ and y = ⎢⎡9.16 ⎥⎤ , respectively)
•
ceiling of –7.98 and ceiling of –4.39 (results should be y = ⎢⎡ −7.98⎥⎤ and y = ⎢⎡ −4.39 ⎥⎤ , respectively)
•
ceiling of
•
respectively)
ceiling of 8 and the ceiling of 6 (results should be y = ⎡⎢8⎤⎥ and y = ⎡⎢ 6 ⎤⎥ , respectively)
39.6 and ceiling of
32.05 (results should be y = ⎡⎢ 39.6 ⎤⎥ and y = ⎡⎢ 32.05 ⎤⎥ ,
Have the class work in pairs on Exercises 1 and 2. Have students check their results with one or more other
pairs. Bring the class together and have pairs share their results on the board or overhead projector using the
calculator view screen. While the students are working, circulate to provide assistance and to clarify questions.
Bring the class together after the first two exercises have been completed. In the second part of this activity,
students will investigate real–world step functions. Remind the class that to expand a “run” by a certain
amount, divide x by that amount inside our step function symbol. To increase the “rise” by a certain amount,
multiply by that amount outside of the step function symbol. Remind the class about the “rise” and “run” that
was discussed in the Setting the Stage activity. We’ll use the same terminology for the example below. The
terminology is used to assist the students in their understanding of the ceiling function and is not necessarily
the standard terminology used when discussing the ceiling function.
Let's take a look at an example involving the pay for a job. Have the class work in pairs. Tell the class to
assume they have a part–time job after school making $10 per hour and that they work 10 hours each week.
Tell the class to assume you have to work a full hour to get any money. Ask the class to create a table of values
that represent hours worked and pay in dollars for their part–time job on the blank side of their dry–erase
board. Have a volunteer share their data table with the class on the board or on a blank transparency on the
overhead projector. Ask the class, "What is the rise and run of your part–time job?" The class should agree that
the rise is $10 and the run is one hour. Now have the class write a step function equation for their job and
remind them they have to work a full hour for payment for each hour. Ask the class, "Will the step function be
a floor or ceiling? Explain." The class should agree that the function is a floor step function since they must
work a full hour to get any pay. Ask for students to share the step function equation they wrote. The class
should agree that the step function is y = 10 ⎣⎢ x ⎦⎥ . Have the student pairs graph their step function on the grid
while a volunteer graphs their step function on a grid transparency. Tell the class to set a scale of 10 for the
vertical axis, y axis. This means that each vertical tick mark equals $10.
AIIF
Non–Linear Functions
Lesson 7: Step Functions
Page 145
y
2300
x (hours worked)
[0, 1)
[1, 2)
[2, 3)
[3, 4)
[4, 5)
[5, 6)
[6, 7)
2200
y value (Pay
in dollars)
0
10
20
30
40
50
60
2100
2000
1900
1800
1700
1600
1500
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
200
100
-5
-4
-3
-2
x
-1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Note: Each tick mark on the x–axis represents 1 unit
and each tick mark on the y–axis represents 10 units.
Now ask the class, "What if we received pay for each 30 minute interval we worked. Would the run be the
same? Would the run be expanded or contracted? Would the rise be the same? What would the new “rise” and
“run” values be for the job?" Have students share their thoughts on the questions. Students should understand
that the “run” and the “rise” will be different because we are earning the same amount per hour but we are
getting half the pay every 30 minutes. Have the class write a new step function equation based on their new
rise and run values and ask for students to share their new step function. The class should agree on the new
step function, y = 5 ⎣⎢ 2 x ⎦⎥ . Have the students graph their new step function and ask them what they should use
for the x and y scales or what units should they label on the axes. Have a volunteer graph the step function
equation on a grid transparency.
y
2300
2200
x (hours worked)
[0, 0.5)
[0.5, 1)
[1, 1.5)
[1.5, 2)
[2, 2.5)
[2.5, 3)
[3, 3.5)
2100
y value ($)
0
5
10
15
20
25
30
2000
1900
1800
1700
1600
1500
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
200
100
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Note: Each tick mark on the x–axis represents 0.5 units and each tick mark on the y–axis represents 5 units.
Now have pairs group with another pair to form a group of four but remain in their pairs. Have one pair do
the first example problem while the other pair does the second example problem. After the problems have
been completed, tell groups to exchange problems and check each other’s work. The pairs should write and
graph a new step function based on the examples below.
Examples
• Your pay is based on 12 minute work intervals
• Your pay is based on 15 minute work intervals
Have pairs share their results with the class. The class should agree that the step function equation for the first
example is y = 2 ⎢⎣ 5x ⎥⎦ and that the equation for the second example is y = 2.5 ⎢⎣ 4 x ⎥⎦ .
AIIF
Page 146
x (hours worked)
[0, 0.2)
[0.2, 0.4)
[0.4, 0.6)
[0.6, 0.8)
[0.8, 1)
[1, 1.2)
[1.2, 1.4)
y value ($)
0
2
4
6
8
10
12
x (hours worked)
[0, 0.25)
[0.25, 0.5)
[0.5, 0.75)
[0.75, 1)
[1, 1.25)
[1.25, 1.5)
[1.5, 1.75)
y
y
2300
2300
2200
2200
2100
2100
2000
2000
1900
1900
1800
1800
1700
1700
1600
1600
1500
1500
1400
1400
1300
1300
1200
1200
1100
1100
1000
1000
900
900
800
800
700
700
600
600
500
500
400
400
300
300
200
200
100
-5
-4
-3
-2
-1
y value ($)
0
2.5
5
7.5
10
12.5
15
100
x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Note: For the first graph each tick mark on the x–axis represents 0.2 units and each tick mark on the y–axis
represents 2 units. For the second graph each tick mark on the x–axis represents 0.25 units and each tick mark
on the y–axis represents 2.5 units.
Working with their partner, have the class work on Exercises 3 through 6. Have pairs check their results with
one or two other pairs. If students are having trouble, have other student partnerships share their results.
AIIF
Non–Linear Functions
Lesson 7: Step Functions
Page 147
Activity 2
SJ Page 69
In this activity, you will investigate evaluating another form of step function, the ceiling function, using your
graphing calculator and the int( ) function. Mathematically, the ceiling of x is called the least integer function.
It represents the smallest integer not less than x. Remember from Activity 1, the int(x) function, mathematically
the floor function, is the greatest integer less than or equal to x. If x = 3.14 then the int(3.14) = 3. However, the
ceiling of 3.14, or ⎡⎢ 3,14 ⎤⎥ , equals 4.
1.
2.
Evaluate the ceiling function using your graphing calculator. Write the function down as it was entered
in the calculator.
a.
⎡⎢ 35.001⎤⎥
The ceiling of 35.001 is 36. I entered int(35.001) + 1 into the graphing calculator.
b.
–4• ⎡⎢ −6.43 ⎤⎥
Negative four times the ceiling of –6.43 is 24. I entered –4(int(–6.43) + 1) into the graphing
calculator.
c.
8• ⎢⎡ 0.045 ⎥⎤
Eight times the ceiling of 0.045 is 8. I entered 8(int(0.045) + 1) into the graphing calculator.
d.
3.5• ⎢⎡ −7.89 ⎥⎤ – 2
Three point five times the ceiling of –7.89 minus two is –26.5 . I entered 3.5(int(–7.89)+1)–2 into
the graphing calculator.
e.
⎡⎢ 4.28 ⎤⎥
The cube of the ceiling of 4.28 is 125. I entered (int(4.28) + 1 )^ 3 into the graphing calculator.
3
Check the results from the ceiling functions below. If any of the results are incorrect, give the correct
result and state what may have caused the incorrect results.
a.
⎡⎢ 23.15 ⎤⎥ = 23 The correct results should be 24. The person may have used the floor function
instead of the ceiling function.
b.
⎢⎡ −7.45 ⎥⎤ = −8 The correct results should be –7. The person may have used the floor function
instead of the ceiling function.
c.
−4 ⎢⎡ 34.678 ⎥⎤ = 140 The correct results should be –140. The person may have multiplied by +4
instead of –4.
d.
⎢⎡ 5.65 ⎥⎤ + 2 = 4 The correct results should be 8. The person may have subtracted 2 instead of
adding 2.
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Page 148
SJ Page 70
In this part of the activity, you will investigate real–world
Note: To expand a “run” by a certain
applications of step functions. In Activity 1, the “run” was
amount, we divide x by that amount
expanded by dividing x by a certain amount inside the step
inside our step function symbol. To
function and contracted (shortened) by multiplying x by a
increase the “rise” by a certain amount,
certain amount inside the step function. Also, the “run” was
we multiply by that amount outside of
increased by multiplying by a certain amount outside the step
the step function symbol.
function and decreased by dividing by a certain amount outside
the step function. For example, in Exercise 2 of Activity 1, if your
sister was paid every half hour, you had to multiply x inside the floor function by 2 to decrease the run, y =
floor (2x). Also, when your sister wanted $2.50 for each hour of work, you had to multiply the floor function
by 2.5, y = 2.5 floor (x).
3.
You have recently graduated from college and have taken a job with a company. Your starting salary is
$30,000 per year. The company pays its employees once a month.
a.
Write a step function equation based on the information in the exercise.
Answers may vary. A sample response is: "The step function equation is y = 2500 ⎢⎣ x ⎥⎦ , where x
represents the month worked.
b.
Create a table of values for an appropriate x interval.
x (months)
[0, 1)
[1, 2)
[2, 3)
[3, 4)
#
[11, 12)
[12, 13)
y value ($)
0
2500
5000
7500
#
27,500
30,000
y
c.
Graph your step function equation.
25,000
20,000
15,000
10,000
5,000
x
1
2
3
4
5
6
7
Months
8
9
10 11 12
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Non–Linear Functions
Lesson 7: Step Functions
Page 149
4.
SJ Page 71
From Exercise 1, the company has decided to pay its employees weekly. Note: Graph a portion of your
function.
a.
Write a step function equation based on the information in the exercise. Note: There are 52 weeks
in a year.
Answers may vary. A sample response is: "The step function equation is y = 576.92 ⎣⎢ x ⎦⎥ , where x
represents the week worked.
b.
Create a table of values for an appropriate x interval.
x (weeks)
[0, 1)
[1, 2)
[2, 3)
[3, 4)
#
[51, 52)
[52, 53)
5.
y value ($)
0
576.92
1153.85
1730.76
#
29,423.08
30,000
y
c.
Graph your step function equation.
15,000
d.
Do you think there are any weeks where
the pay could be different? Explain.
There could be weeks where the pay is one
cent more due to rounding to the nearest
cent. Final pay must equal a whole year's
salary.
10,000
5,000
x
10
Weeks
20
After graduating from college with a degree in meteorology, you have taken up a position with
NOAA, the National Oceanic and Atmospheric Administration. Your first assignment is to introduce a
new tornado scale to replace the current Fujita Scale table shown below. Due to temperature changes
over the past several decades, NOAA has decided to make a more consistent range of wind values for
tornados. A gale force tornado will now start at 50 miles per hour (mph) and
the new scale will have increments of 50 mph. The scale will still go from F0
through F6. Note: This problem deals with a hypothetical situation.
Wind Speed
(MPH)
40–72
73–112
113–157
158–206
207–260
261–318
319–379
The Fujita Scale
F–Scale Number Tornado Classification
F0
F1
F2
F3
F4
F5
F6
Gale tornado
Moderate tornado
Significant tornado
Severe tornado
Devastating tornado
Incredible tornado
Inconceivable tornado
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Page 150
b.
Write a step function equation based on the information in the exercise.
Answers may vary. A sample response is: "The step function equation is y = ⎢⎣ x / 50 ⎥⎦ − 1 , where x
represents the wind speed starting at 50 and going through 400, and y represents the scale from 0
y
through 6.”
Create a table of values for an
appropriate x interval.
x value (wind
speed in mph)
[50, 100)
[100, 150)
[150, 200)
[200, 250)
[250, 300)
[300, 350)
[350,400 )
y (scale)
0
1
2
3
4
5
6
6
5
4
Scale
SJ Page 72
a.
3
2
1
64
192
256
Wind Speed in MPH
320
384
x
c.
Graph your step function equation.
d.
What is the name you have given to your new tornado scale? Write the name of the new scale in
the table above. The name of the new tornado scale will vary. Students may keep the current
name or use their own name for the new tornado scale.
Computers store data using the binary number system, which has only two values, 0 and 1.
Computers use voltages to record data as a 0 or a 1. Low voltages represent a 0 and high voltages
represent a 1. Some computers use a RISC (Reduced Instruction Set Computer) microcontroller which
operates in the range of voltages 0 to 18 volts. Assume that half the voltages represent a 0 (low
voltages) and the other half of the voltages (high voltages) represent a 1.
a.
What is the interval of voltage values that
would represent a binary value of 0 for a
RISC based computer? The interval for low
voltages would be [0, 9).
y
1
b.
What is the interval of voltage values that
would represent a binary value of 1 for a
RISC based computer? The interval for
high voltage would be [9, 18).
c.
Write and graph a step function representing
the two binary values of 0 and 1 for the range
of voltages. The step function would be y = ⎢⎣ x / 9 ⎥⎦ .
Binary Value
6.
128
9
Volts
18
x
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Non–Linear Functions
Lesson 7: Step Functions
Page 151
Practice Exercises
SJ Page 73
For Exercises 1 through 4, evaluate the step function and state the type of step function, floor or ceiling.
1.
y = ⎢⎣ 5.8 ⎥⎦ / 2
2.
y = ⎢⎡ −2.567 ⎥⎤
3.
y = 5 ⎣⎢14.689 ⎦⎥ + 4
The value of the floor step function is 74.
4.
y = ⎡⎢ 37.89 ⎤⎥
The value of the ceiling step function is 7.
The value of the floor step function is 2.5.
3
The value of the ceiling step function is –8.
For Exercises 5 through 7, create a table of appropriate x- and y–values based on the “run” and “rise” for the
given step functions and graph the step function. Pick appropriate scales for your axes.
5.
y = ⎢⎣ x / 5 ⎥⎦
x
[-15, –10)
[–10, –5)
[–5, 0)
[0, 5)
[5, 10)
[10, 15)
y
–3
–2
–1
0
1
2
The graph is a floor step function.
6.
y = ⎣⎢ 5x ⎦⎥
x
[–1.0, –0.8)
[–0.8, –0.6)
[–0.6, –0.4)
[–0.4, –0.2)
[–0.2, 0)
[0, 0.2)
[0.2, 0.4)
[0.4, 0.6)
[0.6, 0.8)
[0.8, 1)
y
–5
–4
–3
–2
–1
0
1
2
3
4
The graph is a floor step function.
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Page 152
SJ Page 74
y = 6 ⎢⎣ x / 3 ⎥⎦
7.
x
[–12, –9)
[–9, –6)
[–6, –3)
[–3, 0)
[0, 3)
[3, 6)
[6, 9)
[9, 12)
y
–24
–18
–12
–6
0
6
12
18
The graph is of a floor step function.
For Exercises 8 and 9, create a table of appropriate x- and y-values for the given graph, write a step function,
and state the type of step function, ceiling or floor.
8.
x
[–4, –3)
[–3, –2)
[–2, –1)
[–1, 0)
[0, 1)
[1, 2)
[2, 3)
(3, 4)
The step function is y = −3 ⎣⎢ x ⎦⎥ − 3 and it is a floor function.
y
9
6
3
0
–3
–6
–9
–12
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Non–Linear Functions
Lesson 7: Step Functions
Page 153
SJ Page 75
9.
x
[–12, –9)
[–9, –6)
[–6, –3)
[–3, 0)
[0, 3)
[3, 6)
[6, 9)
y
–12
–8
–4
0
4
8
12
The step function is y = 4 ⎣⎢ x / 3 ⎥⎦ + 4 and it is of
type floor.
For Exercises 10 and 11:
a. Write a step function equation based on the information in the exercise.
b. Create a table of values for an appropriate x interval.
c. Graph your step function equation.
10.
You have a part–time job after school making $12.00 per hour. Your boss gives you partial pay for every
6 minutes that you work.
a.
Answers may vary. A sample response might be: "The step function is y = 1.2 ⎣⎢ x / 6 ⎦⎥ .
b.
Possible table of values:
x
[0, 6)
[6, 12)
[12, 18)
[18, 24)
[24, 30)
[30, 36)
[36, 42)
[42, 48)
[48, 54)
[54, 60)
[60,66)
y ($)
0.00
1.20
2.40
3.60
4.80
6.00
7.20
8.40
9.60
10.80
12.00
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Page 154
SJ Page 76
11.
A company pays its employees a salary based on the number of years of employment with the company.
New employees start with a salary of $25,000 a year. The company increases the employee’s salary by
$4,000.00 for each completed year of employment.
a.
Answers may vary. A sample response might be: "The step function is y = 4000 ⎢⎣ x ⎥⎦ + 25000 .
b.
Possible table of values:
x (Years)
[0, 1)
[1, 2)
[2, 3)
[3, 4)
[4, 5)
[5, 6)
y (Salary in
dollars)
25,000
29,000
33,000
37,000
41,000
44,000
Non–Linear Functions
Lesson 7: Step Functions
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Page 155
Outcome Sentences
A step function is
A floor step function is
A ceiling step function is
To increase the run of a step function
To increase the rise of a step function
I still need help with
SJ Page 77
AIIF
Page 156
Teacher Reference
y
Lesson 7 Quiz Answers
1.
Answers may vary. A sample table might be:
x
[–24, –16)
[–16, –8)
[–8, 0)
[0, 8)
[8, 16)
[16, 24)
[24, 32)
y
–12
–8
–4
0
4
8
12
8
4
–16
–8
8
16
x
–4
–8
The graph is a floor function.
y
2.
Answers may vary. A sample table might be:
x
[–1.5, –1)
[–1, –0.5)
[–0.5, 0)
[0, 0.5)
[0.5, 1)
[1, 1.5)
[1.5, 2)
y
–1.5
–1
–0.5
0
0.5
1
1.5
The graph is a floor function.
3.
The step function equation is y = 3 ⎢⎣ x / 5 ⎥⎦ . Table of
values is:
x
[–20, –15)
[–15, –10)
[–10, –5)
[–5, 0)
[0, 5)
[5, 10)
[10, 15)
[15, 20)
y
–12
–9
–6
–3
0
3
6
9
1
1
–1
–1
x
AIIF
Non–Linear Functions
Lesson 7: Step Functions
Page 157
Lesson 7 Quiz
Name:
For Questions 1 and 2, create a table of appropriate x- and y-values based on the run and rise for the given step
functions, graph the step function, and state the type of step function, ceiling or floor. Place correct labels and
units on the graph.
1.
y = 4 ⎢⎣ x / 8 ⎥⎦
2.
y = 1 / 2 ⎣⎢ 2 x ⎦⎥
3.
Create a table of appropriate x and y values for the given graph, write a step function, and state the type
of step function, ceiling or floor.
y
10
5
–10
–5
5
–5
–10
10
x
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Page 158
Lesson 8: Miscellaneous Non–Linear Functions
Objectives
• Students will be able to create a table of values for absolute value functions and circle equations
• Students will be able to determine vertical and horizontal translations of absolute value functions and
circle equations
• Students will be able to write equations involving absolute value functions and circle equations
• Students will be able to graph absolute value functions and circle equations
• Students will be able to solve absolute value, piece–wise, and circle equations which apply to real–world
applications
• Students will be able to write and graph piece–wise functions involving real–world applications
Essential Questions
• How do absolute value functions, piece–wise functions, and circle equations apply to real–world
applications?
Tools
• Student journal
• Setting the Stage transparency
• Dry–erase boards, markers, erasers
• Colored pencils
• Graphing calculator and view screen
• Activity 2 transparency
• Activity 4 transparencies
Warm Up
• Problems of the Day
Number of Days for Lesson
• 3 Days (A suggestion is to complete Activity 1 and Practice Exercises 1 through 6 on the first day, Activity
2 and Practice Exercise 7 though 10 on the second day, and then complete Activity 3 and 4 and remaining
Practice Exercises and quiz on the third day.)
Vocabulary
Absolute value
Transformation
Translation
Dilation
Expansion
Contraction
Vertical line test
Equation of a circle
Piece-wise function
Non–Linear Functions
Lesson 8: Miscellaneous Non-Linear Functions
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Page 159
Teacher Reference
Setting the Stage
Divide the class into pairs. Make sure each pair has a dry–erase board. Ask the students if they remember what
absolute value means. See if the class can give some examples of absolute value. The key here is that students
remember that absolute value represents the distance a number is from 0 on a number line and that distance is
always a positive quantity.
Display the Setting the Stage transparency. Ask for a volunteer to complete the table of ordered pairs, remind
the class that they are finding the absolute value of x, and to create a scatter plot of the ordered pairs while the
class does the same in their pairs. The class can assist the volunteer as needed. Ask the students what they
notice about the shape of the scatter plot. Things they could say are, "The shape is like the letter V." Another
should be, “The left half represents a line with negative slope while the right half is a line with positive slope.”
They may also say, “The left half is a mirror image of the right half (or vice versa) and the graph is symmetrical
about the y-axis.” A last possibility, “The graph represents the graph of an even function.” See if the class
could write an equation for the table of ordered pairs. The class might be able to come up with y =|x|.
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Page 160
Setting the Stage
Transparency
Absolute Value
x
–11
–9
–7
–5
–3
–1
y
x
1
3
5
7
9
11
y
Non–Linear Functions
Lesson 8: Miscellaneous Non-Linear Functions
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Page 161
Teacher Reference
Activity 1
Have the class continue to work in pairs. In this activity, the class will investigate absolute value functions and
translations for absolute functions. Ask the class, "We have investigated vertical and horizontal translations,
from the origin, for other functions. How might we translate, vertically and horizontally, absolute value
functions such as y =|x|?" Have student pairs get with another student pair to form a group of four. Have the
groups discuss the question for two to three minutes and create a list of ideas and suggestions to the question.
Have groups share their ideas with the class. Make sure all ideas have been presented. Have a student record
the ideas and suggestions from the groups on the board or on a blank transparency on the overhead projector.
Have the groups who give suggestions also give an example of their suggestions that they would model on a
blank transparency grid on the overhead projector while the class models along on their dry–erase board.
Make sure the volunteers create a table of ordered pairs before drawing their graphs (basically they are
creating a scatter plot of their suggested absolute value equation). Discuss the terms dilation, contraction and
expansion with the class and how it relates to the graph of the absolute value function y = |x|. Students might
be familiar with dilations from transformations in geometry. Use the examples y = |2x| and y = |x/2|. The
first example will contract the graph of y = |x| while the second example will expand the graph of y = |x|.
Discuss which ideas and suggestions translated the absolute value function y = |x| and which ones didn't.
Have the class offer their interpretations of why some worked and why others may not have worked. Tell the
class that the calculator uses the abs() function. This function can be found by pressing the following key
sequence:  ~ Í ( or À). The function abs( will appear. Have the class investigate the responses
written on the board using their graphing calculator. Have student volunteers share the results of their
investigations using the view screen graphing calculator. The following screen shots show how to obtain the
abs( function on the TI–83 or 84 Plus™ graphing calculator.
Have the class work in pairs on Exercises 1 through 17. Have pairs share their results with the rest of the class.
Discuss general forms for translating an absolute value function vertically and horizontally as well as
contracting and expanding absolute value functions.
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Page 162
SJ Page 78
Activity 1
In the modeled exercises, you attempted to determine how to translate, vertically and horizontally, the
absolute function y = |x|. In these exercises, you will continue your investigation of translations of y = |x|
using the graphing calculator.
For Exercises 1 through 8, you will investigate the transformations of the graphs of the absolute value function
from the origin by adding, subtracting, multiplying, and dividing inside and outside the absolute value
brackets. Using your graphing calculator, write an absolute value function for each absolute value situation,
create a table of ordered pairs, and draw the graph on the provided grid. State the type and value of the
transformation on the graph; vertical translation or horizontal translation compared to the graph of y = |x|.
Also state if the graph has been dilated (contracted or expanded) compared to y = |x|. The first exercise has
been completed for you.
1.
Add 5 inside the absolute value brackets.
The function is y = |x + 5|; the graph has been
horizontally translated 5 units to the left. Grid scale is
one horizontal and one vertical unit.
x
–9
–7
–5
–3
–1
1
3
y
y
4
2
0
2
4
6
8
x
2.
Add 5 outside the absolute value brackets.
x
–6
–4
–2
0
2
4
6
y
11
9
7
5
7
9
11
The function is y = |x| + 5; the graph has been
vertically translated 5 units.
Non–Linear Functions
Lesson 8: Miscellaneous Non-Linear Functions
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Page 163
3.
Subtract 3 inside the absolute value brackets.
x
–6
–4
–2
0
2
4
6
y
9
7
5
3
1
1
3
The function is y = |x – 3|; the graph has been
horizontally translated 3 units to the right from
the origin.
4.
Subtract 3 outside the absolute value brackets.
x
–6
–4
–2
0
2
4
6
y
3
1
–1
–3
–1
1
3
The function is y = |x| – 3; the graph has been
vertically translated down by 3 units from the
origin.
5.
Multiply by 2 inside the absolute value brackets.
x
–5
–3
–1
0
1
3
5
y
10
6
2
0
2
6
10
The functions is y = |2x|; y–values have been
contracted by a factor of 2.
SJ Page 79
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Page 164
SJ Page 80
6.
Multiply by 2 outside the absolute value brackets.
x
–5
–3
–1
0
1
3
5
y
10
6
2
0
2
6
10
The functions is y = 2|x|; y–values have been
contracted by a factor of 2.
7.
Multiply by –2 inside the absolute value brackets.
x
–5
–3
–1
0
1
3
5
y
10
6
2
0
2
6
10
The functions is y = |–2x|; y–values have been
contracted by a factor of 2.
8.
Multiply by –2 outside the absolute value brackets.
x
Y
–5
–3
–1
0
1
3
5
–10
–6
–2
0
–2
–6
–10
The functions is y = –2|x|; y–values have been
contracted by a factor of 2.
Non–Linear Functions
Lesson 8: Miscellaneous Non-Linear Functions
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Page 165
9.
SJ Page 81
Was there a difference between Exercises 5 and 6? Explain your answer.
There was no difference between Exercises 5 and 6 because we multiplied by a positive constant which
did not affect the final results.
10.
Was there a difference between Exercises 7 and 8? Explain your answer.
There was a difference between Exercises 7 and 8 because we multiplied by a negative constant which
reflected the graph about the x-axis when multiplied on the outside of the absolute value bracket.
11.
What would you expect the results to be if we divided inside the absolute value brackets and outside
the absolute value brackets by a positive constant? Explain your answer.
Answers may vary. A sample response might be: "There would be no difference because we are dividing
by a positive constant just like when we multiplied by a positive constant."
12.
What would you expect the results to be if we divided inside the absolute value brackets and outside
the absolute value brackets by a negative constant? Explain your answer.
Answers may vary. A sample response might be: "There would be a difference because we would be
dividing by a negative constant which would reflect the graph about the x-axis just like it did when
we multiplied by a negative constant."
13.
When dividing an absolute value function, inside or outside of the absolute value brackets, by a
positive constant, what type of transformation would you expect on the graph: horizontal translation,
vertical translation, dilation (contraction or expansion)? Explain your answer.
Answers may vary. A sample response might be: "The type of transformation would be expansion by
dividing by a positive constant because the y-values have been decreased by a factor of the positive
constant we used to divide by."
For Exercises 14 and 15, write a function for the given situation and draw its graph on the grid provided.
14.
The function y = |x| has been horizontally translated left by 2 units and vertically translated up by 2
units from the origin.
The function is y = |x + 2| + 2.
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Page 166
SJ Page 82
15.
The function y = |x| has been expanded by a factor of 2 and vertically translated down by 4 units from
the origin.
The function is y = |x/ 2| – 4.
For Exercises 16 and 17, write an absolute value function from the given graph
16.
17.
The function for Exercise 16 is y = |x – 4|+5; the function for Exercise 17 is y = |x + 5|–3.
Non–Linear Functions
Lesson 8: Miscellaneous Non-Linear Functions
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Page 167
Teacher Reference
Activity 2
In this activity, students will investigate the equation of a circle and learn how to restrict its domain in order to
make a function. Students will also be able to solve the circle equation for y in order to use the graphing
calculator. Review with students the definition of a function and the Vertical Line Test. Display Activity 2
transparency. Divide the class into groups of four using your favorite grouping strategy. Have the class
discuss in their groups which shapes represent a function and which shapes do not represent a function. Tell
the class to be prepared to explain their results. Give the groups 3 to 5 minutes for discussion. Have groups
volunteer their results to the class. Each group should give their results for at least one shape.
Ask the class, "How might we restrict the range for the non–function graphs so that the graph is the graph of a
function?" You may need to review the term range. Students can either discuss the question in their groups
and then share results or you can lead a class discussion on the question. Have a volunteer record the class'
responses on the board or on the activity transparency. Make sure the class agrees on how to restrict the range
of each non–function graph to make it the graph of a function. This concept is important so that the class can
understand that we can make the graph of a circle a function if we restrict its range.
Tell the class that the equation x 2 + y 2 = 100 has certain values that make it true. Using graphing calculators,
have the students, in their groups, find integer values for x and y, that make the equation true. Ask groups to
share one ordered pair of integer values that makes the equation true. Have a student list the group responses
on the board or on a blank transparency on the overhead. Continue to ask groups to share their integer value
ordered pairs until all ordered pairs from the table below have been shared.
x
y
Have the groups plot the ordered pairs on a dry–erase board and draw a connected
–10
0
graph as smooth as possible. Ask the class, "What do the connected ordered pairs
–8
–6
form?" The students should realize that the connected ordered pairs form a circle.
–8
6
Ask the class, "What is the center of your circle?" and "How far from the center of
–6
–8
the circle is each point on the circle?" These questions are to gauge the student's
–6
8
prior understanding of circles. This should include the center, the radius, and the
0
–10
fact a circle represents all the points that are equidistant from a point known as the
0
10
center of the circle.
6
8
6
–8
Ask the students, "How could we rewrite our equation so we could enter it into the
8
6
graphing calculator to graph?" and "What format must the equation be in so that we
8
–6
can enter it into the graphing calculator?" The goal is for students to understand
10
0
that they must rewrite the equation into two separate equations
y = 100 − x 2 and y = − 100 − x 2 . You may need to assist students in rewriting the equations by reminding
them of their equation solving skills from previous units. Have the class enter these equations into the
graphing calculator to graph the circle. Let the class know that because the grid on the graphing calculator is
not a square, the graph may not look circular. Another way to enter the two equations, at least for a TI–83 or 84
Plus™ graphing calculator, would be to enter the first equation in Y1 and then enter –Y1 into Y2.
Now, ask the class, "We have seen many ways to translate functions from the origin. We know how to
translate the graph of a parabola and power functions. In the previous activity we did transformations of
absolute value functions. How might we write our equation of a circle to translate it horizontally and/or
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vertically from the origin?" You might want to give an example for a parabola (quadratic function)
2
like y = x 2 and y = ( x − 4 ) to show a horizontal translation and y = x 2 + 5 to show a vertical translation. Have
the students discuss and investigate in their groups how to translate a circle from the origin. Have the class
work with the equation x 2 + y 2 = 25 . Tell the class they can either work on the equation with or without a
graphing calculator. If they use a graphing calculator, let them know that they could just investigate "half" a
circle by using the equation y = 25 − x 2 . Ask groups to share their results with the class. Have a student list
the various methods shared by groups on the board or on the transparency used earlier. Make sure that
information on the center of the circle after it has been translated was presented. If not, let the class know what
the coordinates of a circle are after it has been translated.
2
2
Write the general form of the equation of a circle on the board or blank transparency: ( x − h ) + ( y − k ) = r 2 .
Tell the class that the coordinates (h, k) not only represents the horizontal and vertical translations, but also the
center of the circle and that when the origin is the center of the circle we get the equation x 2 + y 2 = r 2 .
Ask, "How many intercepts do you think a circle with its center at the origin has? Explain." The students could
use their dry–erase boards to assist them in answering the question. The key is that students understand that
there are four intercepts and this concept can be used to draw a rough sketch of any equation of a circle
centered at the origin. You could also ask, "How can the center of the circle and the circle's radius be used to
help us plot points to graph the equation of a circle?" The center and radius can be used to plot four points that
could be used as the cornerstone points to draw the graph of a circle. Have the class investigate graphing
equations of a circle in their groups. Model, or have a student model, graphing the equation x 2 + y 2 = 16 on a
transparency grid while the class parallels with x 2 + y 2 = 9 on their dry–erase boards. A simple graphing
strategy could be the following:
• Locate and plot the center of the circle at the origin
• Determine the value of the radius
• From the center (origin) of the circle, go up the number of units equal to the radius and plot a point.
• From the center (origin) of the circle, go down the number of units equal to the radius and plot a point.
• From the center (origin) of the circle, go left the number of units equal to the radius and plot a point.
• From the center (origin) of the circle, go right the number of units equal to the radius and plot a point.
• Draw the circle
Have a volunteer(s) model graphing the first circle equation from the sample exercises below while the class
follows along using the second equation from the sample exercises. The volunteer(s) should also find the
coordinates of the center of the circle and four points associated with the center and radius r and solve their
equation for y.
Sample Exercises
•
x 2 + y 2 = 4 and x 2 + y 2 = 9
•
x 2 + y 2 = 49 and x 2 + y 2 = 25
2
2
Remind the class about the general form of the equation of a circle, ( x − h ) + ( y − k ) = r 2 , with center at (h, k).
Ask the class, "How might we use the previous graphing technique for circles with the center at the origin to
graph a circle with the center at some other location other than the origin?" The goal here is for students to
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Lesson 8: Miscellaneous Non-Linear Functions
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realize that after the center of the circle has been determined and a point plotted for it, they can use the same
technique of going left and right, up and down by the value of the radius to sketch a rough graph of the circle.
Have a volunteer(s) model graphing the first circle equation from the sample exercises below while the class
follows along using the second problem from the sample exercises. The volunteer(s) should also find the
coordinates of the center of the circle and four points associated with the center and radius r to graph the
equation. Also, have the students solve the equation for y.
Sample Exercises
•
•
•
( x − 6 )2 + y 2 = 16 and ( x − 3 )2 + y 2 = 25
2
2
x 2 + ( y − 7 ) = 9 and x 2 + ( y − 5 ) = 4
( x + 3 )2 + ( y + 2 )2 = 36 and ( x + 1)2 + ( y + 3 )2 = 49
A simple graphing strategy could be the following:
• Locate and plot the center of the circle at the origin
• Determine the value of the radius
• From the center of the circle, go up the number of units equal to the radius and plot a point.
• From the center of the circle, go down the number of units equal to the radius and plot a point.
• From the center of the circle, go left the number of units equal to the radius and plot a point.
• From the center of the circle, go right the number of units equal to the radius and plot a point.
• Draw the circle
Have the class work in pairs within their groups of four on Exercises 1 through 13. Have student pairs check
their results with the other student pair in the group and possibly other groups. As you're walking around,
note which groups have a good understanding and which groups need help. Have the groups that understand
the content help the other groups by either going to those groups and assisting or sharing their results at the
front of the class.
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Activity 2 Transparency
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Activity 2
SJ Page 83
In this activity, you will investigate graphing the equations
of a circle.
For Exercises 1 through 6 (draw two circles per grid):
a. State the radius and center of the circle.
b. Draw a rough sketch of the circle equation. Use only
four points to draw the rough sketch of the circle.
State the coordinates of the four points.
1.
2.
3.
4.
x 2 + y 2 = 81
a.
The radius is 9 units and the center is (0, 0).
b.
The four points are (–9, 0), (9, 0), (0, –9), and (0, 9).
x 2 + y 2 = 121
a.
The radius is 11 units and the center is (0, 0).
b.
The four points are (–11, 0), (11, 0), (0, –11), and
(0, 11).
( x − 6 )2 + y 2 = 36
a.
The radius is 6 units and the center is (6, 0).
b.
The four points are (0, 0), (12, 0), (6, –6), and (6, 6).
2
x 2 + ( y + 5 ) = 25
a.
The radius is 5 units and the center is (0, –5).
b.
The four points are (5, –5), (–5, –5), (0, –10), and (0, 0).
General Equation of a Circle with
Center at (0, 0) and Radius r
x2 + y 2 = r 2
General Equation of a Circle with
Center at (h, k) and Radius r
( x − h )2 + ( y − k )2 = r 2
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SJ Page 84
5.
6.
7.
( x + 4 )2 + ( y − 3 )2 = 49
a.
The radius is 7 units and the center is (–4, 3).
b.
The four points are (–11, 3), (3, 3), (–4, 10), and
(–4, –4).
( x − 6 )2 + ( y + 3 )2 = 64
a.
The radius is 8 units and the center is (6, –3).
b.
The four points are (–2, –3), (14, –3), (6, 5), and
(6, –11).
Explain the technique you used to find the coordinates of the four points for circles that had a center at
any location other than (0, 0).
Answers will vary. A sample response might be: "After determining the coordinates for the center, I
added and subtracted the value of the radius from the x–coordinate of the center to get two points on a
line parallel to the x–axis. I then added and subtracted the value of the radius to the y–coordinate of the
center to get two more points on a line parallel to the y–axis.
For Exercise 8, pick any three equations from Exercises 1 through 6 and solve them for y.
8.
Answers will vary. The equations in Exercises 1 through 6 solved for y are:
y = ± 81 − x 2
y = ± 121 − x 2
y = ± 36 − ( x − 6 )
2
y = ± 25 − x 2 − 5
2
y = ± 49 − ( x + 4 ) + 3
2
y = ± 64 − ( x − 6 ) − 3
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SJ Page 85
9.
2
2
Using the general equation of a circle, ( x − h ) + ( y − k ) = r , with center at (h, k) and radius r, solve the
2
equation for y to obtain a function equation for a circle.
2
y = ± r 2 − ( x − h) + k
For Exercises 10 through 13, write the equation, in general form, for the graphed circle. State the center of the
circle and the four points used to define its graph.
10.
11.
The equation is x 2 + y 2 = 144 , the center is
(0, 0) and the four points are (–12, 0), (12, 0),
(0, -12), and (0, 12).
2
2
The equation is ( x + 7 ) + ( y + 7 ) = 49 , the
center is (–7, –7) and the four points are
(–14, –7), (0, –7), (–7, 0), and (–7, –14).
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SJ Page 85 (cont.)
12.
13.
2
2
The equation is ( x − 12 ) + ( y − 11) = 25 , the
center is (12, 11) and the four points are (7, 11),
(17, 11), (12, 16), and (12, 6).
2
2
The equation is ( x + 7 ) + ( y − 7 ) = 100 ,
the center is (–7, 7) and the four points
are (–7, –3), (–7, 17), (3, 7), and (–17, 7).
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Teacher Reference
Activity 3
In this activity, students will investigate piece–wise functions, piece–wise function with real–world
applications, and graphs of piece–wise functions. Have the students continue to work as pairs in their groups
of four. Ask the class, "How do you think a piece–wise functions is constructed?" Students may say, "They are
constructed in pieces." Technically, they are correct. Piece–wise functions have different "definitions," formulas
for different values of the domain. Write the following piece–wise function on the board or on a grid
for x < 0 ⎫
⎧5
transparency on the overhead projector: y = ⎨
⎬ . Have a volunteer graph the piece–wise
⎩ 2 x + 5 for x ≥ 0 ⎭
function on the board or a grid transparency while the class graphs along on their dry–erase boards and assists
the volunteer as needed. This is an example of continuous piece–wise function because the graph is not split or
broken. Have other volunteers model the first of each sample exercises below while the class works on the
second of each exercise below.
Sample Exercises
⎧−x + 5 for x < 0 ⎫
⎧−x + 2 for x < 0 ⎫
• y=⎨
⎬
⎬ and y = ⎨
⎩x − 5 for x ≥ 0 ⎭
⎩x − 2 for x ≥ 0 ⎭
•
⎧⎪ 3x + 6 for x < 0 ⎫⎪
⎧⎪ −4 x + 2 for x < 0 ⎫⎪
y=⎨ 2
⎬ and y = ⎨ 2
⎬
for x ≥ 0 ⎪⎭
for x ≥ 0 ⎭⎪
⎩⎪ x
⎩⎪ − x
•
for x < −4 ⎫
for x < -5
⎧−3
⎫
⎧8
⎪
⎪
⎪
⎪
for − 4 ≤ x < 4 ⎬
y = ⎨x + 1
for − 5 ≤ x < 5 ⎬ and y = ⎨x + 5
⎪−3x + 9 for x ≥ 5
⎪
⎪
⎪
⎩
⎭
⎩−2 x + 12 for x ≥ 4
⎭
Discuss the endpoints. Remind the students how they graphed endpoints for inequalities. Ask, "When do we
have solid endpoints and when do we have hollow endpoints?" The class should remember their graphing
techniques from inequalities, both one and two variable inequalities.
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The graphs of the sample exercises are displayed below.
Non–Linear Functions
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These functions can also be graphed on the graphing calculator. It is important that the equations and the
domain be kept inside separate parentheses when using the graphing calculator. Use the first sample exercise
to demonstrate how to use the graphing calculator to graph piece–wise functions. The instructions below are
based on a TI-83 or 84 Plus™ graphing calculator. If you use a different calculator, consult the owner's manual
for an equivalent function.
The first equation is y = –x + 1 for 0 < x. We would need to enter the equation inside parentheses, (–x + 1), and
the domain inside parentheses as well, (0 < x). Also, remind the class that the inequality symbols are under the
TEST menu. The key sequence is y  then press the appropriate number key for the required inequality
symbol. For the second equation, y = x – 1 for x ≥ 0. We would need to enter the equation inside parentheses, (x
– 1), and the domain inside parentheses as well, (x ≥ 0). Set the graphing parameters to be: –12 for x–minimum,
13 for x–maximum, –12 for y–minimum, 13 for y–maximum, and x- and y-scale values of 1. The screen shots
below show the equations in the Y= editor and the resulting graph. Note: The graphs should be dotted (not
connected). Sometimes a dotted graph is easier to view and sometimes it makes it more difficult to view.
Have the class work in pairs on Exercises 1 through 9. Tell the class they will need to write a piece–wise
equation for Exercises 5 through 7 from a graph and for Exercises 8 and 9 they will have to write and graph a
piece–wise equation from an application problem. Let them know that they have written many different
equations for real–world applications before. For these exercises, they will need to pay particular attention to
the domain values for their piece–wise equations, just like they did when they wrote inequality equations
from real–world application problems. Have volunteers share their results on the board or overhead projector
with the class.
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SJ Page 86
Activity 3
In this activity, you will investigate piece–wise functions and
their graphs. You will also use the graphing calculator to graph
piece–wise equations. Follow your teacher's directions for
graphing piece–wise equations on the classroom graphing
calculator.
For Exercises 1 through 4, graph the piece–wise equation on the grid provided. Also, graph the piece–wise
equation on your graphing calculator and write the format of the equation as it was entered into the graphing
calculator.
1.
⎧ 5x + 3 for x < 0 ⎫
y=⎨
⎬
⎩ −2 x + 7 for x ≥ 0 ⎭
Answers may vary. A sample response might be: "I
entered (5x + 3)(x < 0) in Y1 and (–2x + 7)(x ≥ 0) in Y2."
2.
for x < 0 ⎪⎫
⎪⎧2 x 2
y=⎨
⎬
⎩⎪3x − 6 for x ≥ 0 ⎪⎭
Answers may vary. A sample response might be: "I
entered (2x2)(x < 0) in Y1 and (3x – 6)(x ≥ 0) in Y2."
3.
⎧2 x for x < 2 ⎫
⎪
⎪
y=⎨
⎬
⎪⎩ 8 x for x ≥ 2 ⎪⎭
Answers may vary. A sample response might be: "I
entered (2^x)(x < 2) in Y1 and (8/x)(x ≥ 2) in Y2."
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SJ Page 87
4.
⎧ 4 for x < −2
⎫
⎪ 3
⎪
y = ⎨x for − 2 ≤ x < 2 ⎬
⎪−4 for x ≥ 2
⎪
⎩
⎭
Answers may vary. A sample response might be: "I entered
(4)(x < –2) in Y1, (x3)(–2 ≤ x and x < 2) in Y2, and (–4)(2 ≥ x)
for Y3."
For Exercises 5 and 6, write a piece–wise equation for the given
graph.
5.
⎧ − x + 1 for x < 0 ⎫
The piece–wise equation for the graph is y = ⎨
⎬.
for x > 0 ⎭
⎩x
6.
for x < 2⎫
⎧9
The piece–wise equation for the graph is y = ⎨
⎬.
⎩ − x + 4 for x ≥ 2 ⎭
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SJ Page 88
For Exercises 7 through 9, write a piece–wise equation from the given information and then graph your piece–
wise equation. Set axes scales accordingly.
7.
The a local electric company charges $0.0968
per kilowatt hour (KWH) for the first 200 KWH
used and then $0.0762 per kilowatt hour used
beyond the initial 200 KWH. What does the
value of y represent?
The piece–wise equation for the graph
for x ≤ 200
⎧0.0968x
⎫
is y = ⎨
⎬.
⎩0.0762( x − 200) + 19.36 for x > 200 ⎭
The value of y represents the total cost of
electricity.
20
200
8.
A cell phone company charges a $39.99
monthly fee that includes 500 anytime cell
minutes. If you use more than 500 cell minutes,
the cell phone company charges $0.40 for each
additional minute. What does the value of y
represent?
The piece–wise equation for the graph
for 0 ≤ x < 500 ⎫
⎧ 39.99
is y = ⎨
⎬.
⎩0.40( x - 500) + 39.99 for x ≥ 500
⎭
The value of y represents the total monthly cell
phone bill.
30
200
9.
The Reel Time movie theater charges $4.50 for
children younger than 12 and for adults 65 and
older. Everybody else must pay the full price of
$10.
The piece–wise equation for the graph
⎧ 4.50 for x < 12
⎫
⎪
⎪
is y = ⎨ 10 for 12 ≤ x < 65 ⎬ .
⎪ 4.50 for x ≥ 65
⎪
⎩
⎭
12
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Teacher Reference
Activity 4 Courting the Graphing Calculator (Optional)
This activity is meant to be both fun and challenging. Have the students work as pairs in their group of four.
Students will use their knowledge and understanding of the circle equation, written in y = format, and the
graphing calculator to draw half an NBA size basketball court. Also piece–wise equations will be used. The
dimensions of an NBA basketball court are given in the student's journal.
Display Activity 4 transparency. Point out that this view is from the end of the court to midcourt, the
suggested view that they display on their graphing calculator. Point out the dimensions of the basketball
court. Tell the students that they are only going to draw half the court on their graphing calculator. They can
do the view from the end of the court to midcourt or from midcourt to end of the court.
Tell them that they can use horizontal line equations (y =) to draw the boundary line, midcourt line, and the
free throw line, and that the free throw line will require a piece–wise equation and the inequalities will need to
be written in compound form using AND (LOGIC menu under TEST). For example, the inequality -6 ≤ x ≤ 6
would have to be entered as –6 ≤ x AND x ≤ 6. Lead a discussion on the format of the equations needed to
draw the half circles. Tell the class that some of the circles represent the top half or bottom of a circle. Ask the
class, "What equation format would represent a lower half circle?" The class may remember that
y = r 2 − x 2 represented the upper half of a circle with center at the origin and the opposite, y = − r 2 − x 2 ,
would represent the lower half. Give the class recommendations for the graphing window parameters as
shown in screen shots below. You might want to ask, "Why are we suggesting the x graphing window
parameters range from –25 to 25, but the y graphing window parameters only have positive values?" The key
concept is that writing the half circle equations is easier if we only need a vertical translation instead of both a
vertical and horizontal translation in our equations. It also gives the students a line of symmetry to use.
For the students to draw vertical lines they will need to use the DRAW menu (y <). Tell them to use the
second option, Line(, and not the fourth option, Vertical, because Vertical will draw a vertical line the size of
the calculator screen and they won't be able to control its length. The screen shots below show how to use the
Line( option. Tell them that they can draw all of their vertical lines at one time. Also, it is important to tell the
class to be in "graph" mode, meaning to press the s key before y <. To obtain the blinking + cursor
to start drawing a line press the key sequence y < Á. Use the left, right, up, and down arrow keys to
position the + cursor at the location where the line will start. Press the Í key to mark the location. The
blinking + cursor will then change to a blinking rectangle cursor. Then use the up or down arrow key to move
the rectangle cursor to the location for the end of the line and then press the Í key. Continue in such a
manner to draw all the vertical lines at the same time. After the vertical lines have been drawn tell the class to
press the s key to terminate DRAW mode. Have the class practice drawing multiple vertical lines before
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tackling the court challenge. Note: Students can use the Line( function from the home screen and enter
endpoints instead of approximating them from the previous technique discussed. The format is Line(X1, Y1,
X2, Y2) where (X1, Y1) and (X2, Y2) are the coordinates of the endpoints.
The following screen shots show the equations in the Y= editor as well as the end result. Tell the students that
because the rim is quite small and very close to the boundary line, they can use the Circle( option, option 9 in
the DRAW menu, to draw a circle for the rim, but they should make it a little larger than it actually is and a
little farther than the boundary line. A circle is drawn in the same manner as a line. Move the blinking + cursor
to the location for the center of the circle, press the Í key to obtain the blinking rectangle cursor, and then
use the arrow keys to position the cursor for the radius of the circle and press the Í key once more. Note:
You can also draw a circle from the home screen using the format: Circle(X,Y,radius) where (X, Y) is the center
of the circle and radius represents the radius of the circle.
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Activity 4 Transparency
three point line
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SJ Page 89
Activity 4: Courting the Graphing Calculator
In this activity, you will use your knowledge and understanding of equations for horizontal lines, half circles,
and piece–wise functions to draw half a basketball court on your graphing calculator. Follow the instructions
given by your teacher to draw vertical lines as needed on your calculator screen. The information and diagram
below show requirements of a basketball court.
The dimensions of an NBA basketball court are:
• Length of court: 94 feet
• Width 50 feet
• Diameter of rim: 18 inches
• Distance from backboard to free throw line: 19 feet
• Distance from backboard (boundary line) to rim: 6 inches
• Width of the key: 12 feet
• Three-point line/arc: From the center of the rim (basketball hoop) to the three-point line is 22.5 feet.
From the center of the rim to the arc is 23.75 feet.
three point line
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Lesson 8: Miscellaneous Non-Linear Functions
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Practice Exercises
SJ Page 90
For each Exercises 1 and 4, create a table of ordered pairs based on the given absolute value function, graph
the absolute value function, and state any horizontal or vertical translation from the origin and whether the
graph has been expanded or contracted. Label the axis and units on the graph
y
1.
y = x+6
x
y
8
–14
4
–10
0
–6
4
–2
2
8
6
12
The graph is translated horizontally to the left by 6 units.
2.
x
5
–5
–5
y
y = x−4 −6
x
y
10
–12
6
–8
2
–4
0
–2
4
–6
8
–2
12
2
The graph is translated horizontally to the right by 4
units and vertically down 6 units.
3.
5
5
–5
y
y = x /3
x
y
3
–9
2
–6
1
–3
0
0
3
1
6
2
9
3
The graph is expanded by a factor of 3.
x
5
–5
5
x
5
–5
–5
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y
SJ Page 91
4.
y =2 x+3 +1
5
x
–7
–5
–3
–1
1
3
5
y
9
5
1
5
9
13
17
x
5
–5
–5
The graph is horizontally translated to the left by 3 units, vertically up by 1 unit, and contracted by a
factor of 2.
For Exercises 5 and 6, create a table of ordered pairs for the given graphs, write an absolute value equation,
and state any horizontal or vertical translation from the origin and whether the graph has been expanded or
contracted.
y
5.
5
5
–5
–5
x
x
–6
–4
–2
0
2
4
6
8
y
7
5
3
1
–1
–3
–1
1
The absolute value equation is y = |x – 4| – 3; horizontal translation of 4 units to the right and a
vertical translation of –3 units.
y
6.
x
–8
–6
–4
–2
0
2
4
6
y
5
4
3
2
1
2
3
4
5
x
–5
5
–5
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Lesson 8: Miscellaneous Non-Linear Functions
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The absolute value equation is y = |x/2| + 1; vertical translation of 1 unit, expanded by a factor of 2.
For Exercises 7 through 10 (draw two circles per grid):
SJ Page 92
a. State the radius and center of the circle.
b. Draw a rough sketch of the circle represented by the equation. Use only four points to draw the rough
sketch of the circle. State the coordinates of the four points.
c. Solve the equation for y.
y
7.
x2 + y 2 = 4 .
a. The radius of the circle is 2.
b. The four points for the rough sketch of the graph are
(–2, 0), (2, 0), (0, 2), and (0, –2); the center of the circle
is (0, 0).
c.
5
y = ± 4 − x2
x
–5
8.
( x + 4 )2 + y 2 = 64 .
–5
a. The radius of the circle is 8.
b. The four points for the rough sketch of the graph are
(–12, 0), (4, 0), (–4, 8), and (–4, –8); the center of the
circle is (–4, 0).
c.
9.
y = ± 64 − ( x + 4 )
5
2
( x − 6 )2 + ( y + 4 )2 = 25 .
y
a. The radius of the circle is 5.
b. The four points for the rough sketch of the graph are
(1, –4), (11, –4), (6, 1), and (6, –9); the center of the
circle is (6, –4).
c.
5
2
y = ± 25 − ( x − 6 ) − 4
x
10.
( x + 1)2 + ( y + 2 )2 = 81 .
a. The radius of the circle is 9.
b. The four points for the rough sketch of the graph are
(–10, –2), (8, –2), (–1, 7), and (–1, –11); the center of the
circle is (–1, –2).
c.
2
y = ± 81 − ( x + 1) − 2
–5
5
–5
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SJ Page 93
For each of the Exercises 11 through 14, graph the given piece–wise function on the provided grid. Also, graph
the piece–wise equation on your graphing calculator and write the format of the equation as it was entered
into the graphing calculator.
11.
for x < 0 ⎫
⎧ −1
y=⎨
⎬
⎩ 3x − 1 for x ≥ 0 ⎭
Answers may vary. A sample response might
be: "I entered (–1)(x<0) in Y1, (3x–1)(x≥0) in Y2."
13.
⎧⎪−( x + 2)2 for x < −2 ⎫⎪
y=⎨
⎬
x
for x ≥ −2 ⎪⎭
⎪⎩ 4(2 )
Answers may vary. A sample response might
be: "I entered (–(x+2)^2)(x<–2) in Y1, (4(2^x))(x≥–2)
in Y2."
12.
⎧|x + 4| for x < −5 ⎫
y=⎨
⎬
⎩ x / 5 for x ≥ −5 ⎭
Answers may vary. A sample response
might be: "I entered (abs(x+4))(x<–5) in Y1,
(x/5)(x≥–5) in Y2."
14.
−2>x ⎫
⎧−x
⎪⎪
⎪⎪
y = ⎨( x − 2)2 − 2 − 2 ≤ x ≤ 2 ⎬
⎪ 3
⎪
2<x
⎩⎪ x
⎭⎪
Answers may vary. A sample response
might be: "I entered (–x))(x < –2) in Y1,
((x–2)^2–2))(–2≤x and x≤2) in Y2, (x^3)(x>2)
in Y3."
AIIF
Non–Linear Functions
Lesson 8: Miscellaneous Non-Linear Functions
Page 189
15.
SJ Page 94
A cell phone company offers broadband wireless internet access at a cost of $50 per month for the first
1 GB (giga byte) of usage. After the first 1 GB of usage, the company charges $0.50 per 1 MB (mega
byte). Write a piece–wise equation representing the total monthly cost for broadband wireless internet.
Note: 1 GB = 1,000 MB. Label the independent
variable and state what it represents.
for 0 ≤ x ≤ 1000
⎧ 50
y=⎨
⎩0.05( x − 1000) + 50 for x > 1000
The independent variable is x and it represents
the amount of usage in megabytes ($0.05 for 1
MB). The dependent variable is y and it
represents the total cost in dollars of
broadband wireless internet usage.
200
150
100
50
400
800
1200
1600
x (MB)
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Page 190
SJ Page 95
Outcome Sentences
Graphing absolute value functions was similar to
When graphing a circle equation on the graphing calculator
When solving a circle equation for y
Piece–wise graphing was hard to understand because
When graphing a piece–wise equation
I would like to know more about
Non–Linear Functions
Lesson 8: Miscellaneous Non-Linear Functions
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Page 191
Teacher Reference
Lesson 8 Quiz Answers
Answers may vary. A sample table might be:
1.
x
–7
–5
–3
–1
1
3
5
y
12
8
4
0
–2
2
6
A sample transformation response might be: "There
is a horizontal translation of 1/2 unit and vertical
translation –3 units. The graph is contracted because
of multiplication of x by 2."
2.
a.
b.
c.
The circle equation in general form is
( x + 3 ) 2 + ( y − 4 ) 2 = 49 .
The radius of the circle is 7 and the center is
(-3, 4).
The coordinates of the four points are (–3, 11),
(–3, –3), (4, 4),and (–10, 4).
3.
Answers may vary. A sample response might be: "I entered ((x–2)^2)(x ≤ 2) in Y1 and (–x +6)(x > 2) in
Y2."
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Page 192
Lesson 8 Quiz
1.
Name:
y = 2 x − 1 − 3 . Create a table of ordered pairs based on the given absolute value function, graph the
absolute value function, and state any horizontal or vertical translation and whether the graph has been
expanded or contracted. Label the axes and units on the graph.
x
y
2. For y = 49 − ( x + 3)2 + 4
a. Rewrite the equation of the circle in general form.
b. State the radius and center of the circle.
c. Draw a rough sketch of the circle equation. Use only
four points to draw the rough sketch of the circle. State
the coordinates of the four points.
⎧⎪( x − 2)2 for x ≤ 2
3. For y = ⎨
, graph the given piece–wise
⎪⎩ − x + 6 for x > 2
function. Also, graph the piece–wise equation on your
graphing calculator and write the format of the equation as
it was entered into the graphing calculator.
AIIF
Non–Linear Functions
Assessment
Page 193
Teacher Reference
Non–Linear Functions Assessment
Many of the exercises are open–ended items. The problems are grouped in sets of four problems each. Have
the students complete two problems from each set. For the last exercise, have all students determine which
type of functions have the characteristics listed. A characteristic may apply to more than one function type.
Answers
1.
Answers will vary. A sample graph might be:
2.
Since the vertex is (3, –2), and
−b
= 3 or b = −6 a . Also,
2a
y = ax 2 − 6 ax + c and c = –5 since the y–intercept is (0, –5).
Using the coordinates of the vertex and the quadratic
function y = ax 2 − 6 ax + c , we have −2 = a(3)2 − 6 a(3) − 5 .
Solving we get that a =
function is y =
3.
−1
and b = -6(–1/3) = 2. Our
3
−1 2
x + 2x − 5 .
3
Answers will vary. A sample function might be: y = x 2 + x + 1 . The discriminant is
b 2 − 4 ac = 12 − 4(1)(1) = 1 − 4 = −3 . Since the discriminant is less than zero, there are no real solutions.
The student could also graph their equation and show that the graph does not intersect the x–axis and
hence has no solutions.
4a.
4b.
4c.
The 2000 represents the fixed costs.
The 250 represents the selling cost of each unit.
The profit function, P(x), is R(x) – C(x) = 250 x − (2000 + 40 x + x 2 ) = 210 x − 2000 − x 2 .
4d.
The break–even points are where R(x) = C(x) or P(x) = 0. Solving 210 x − 2000 − x 2 = 0 we get x = 10 units
and x = 200 units. Therefore, the points are (10, 2500) and
#5
(200, 50000)
The company makes a profit when 10 < x < 200.
The maximum profit is $9,025 for 105 units.
4e.
4f.
5.
Answers will vary. A sample graph might be:
Note: y–axis scale is 2:1
6.
Answers will vary. A sample graph might be:
Note: y–axis scale is 2:1
#6
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Page 194
7.
Answers will vary. A sample response might be: "The equation of a power function is y = x 3 . Rewriting
the equation so that it is reflected about the x–axis is y = − x 3 .
8.
Answers will vary. A sample response might be: "The equation of a power–like function with a vertical
translation of –3 from the origin and a horizontal translation of +4 from the origin is y = ( x − 4)3 − 3
9.
Answers will vary. A sample graph is displayed in the
grid to the right.
10.
Answers will vary. A sample graph is displayed in the
grid to the right. A sample response might be: "The y–
intercept could represent the initial amount for a growth
function."
11.
Answers will vary. A sample response might be: "A
power function has a constant exponent while an
exponential function has a variable exponent. An
example of a power function is y = x 3 and an example of
an exponential function is y = 2 x .
12.
There will be 997.99 pounds of uranium remaining after 70 years.
13.
The constant of variation, k, is 20. t = 1 when p = 20.
14.
The range of a circle must be restricted so that the equation represents a function.
15.
y = ± 25 − ( x − 4)2 − 2 ; the radius of the circle is 5 and the center has coordinates (4, –2).
16.
The equation of an absolute value function that has been horizontally translated –3 units from the
origin and vertically translated +2 units from the origin is y =|x + 3|+2 .
17.
The graph of Problem 17 appears to the right.
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Non–Linear Functions
Assessment
Page 195
18.
Answers will vary. A sample response might be: "A piece–wise function that is linear for x ≤ 0 and
⎧x x ≤ 0
⎪
varies inversely for x > 0 is y = ⎨ 1
⎪⎩ x x > 0
19.
a.
The step function equation is y = 2.50 ⎢⎣ 4 x ⎥⎦ , where x represents the hours worked.
b.
Answers will vary. A sample response might be:
x (hours)
[0, 0.25)
[0.25, 0.5)
[0.5, 0.75)
[0.75, 1)
[1, 1.25)
[1.25, 1.5)
[1.5, 1.75)
[1.75, 2)
[2, 2.25)
y ($)
0
2.50
5
7.50
10
12.50
15
17.50
20
c.
60
50
40
30
20
10
.5
1
1.5 2
2.5
3
3.5 4
4.5
5
5.5
6
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Page 196
20.
Has a
Maximum or
Minimum
Has
Symmetry
N
Can be
Horizontally
and/or
Vertically
Translated
A
A
A
A
N
A
S
S
Exponential
Function
Inverse Function
N
A
A
A
N
A
N
A
S
S
Absolute Value
Function
A
N
A
A
A
Piece–Wise
Function
Circle Function
S
S
N/A
S
S
A
N
A
A
A
Step Function
A
N
A
S
N
Quadratic
Function
Power Function
Has
Constant
Exponent
Has Variable
Exponent
A
Piece–wise function, constant exponent
⎧1
⎪ x<0
example: y = ⎨ x
⎪x 2 x ≥ 0
⎩
Piece–wise function, variable exponent
⎧1
⎪ x<0
example: y = ⎨ x
⎪2 x x ≥ 0
⎩
Power function maximum or minimum
example: y = x 4
Inverse function maximum or minimum
1
example: y = 2
x
Power function symmetry example: y = x 4
Inverse function symmetry example: y =
Piece–wise function, symmetry
⎧ −x x > 0
⎪
example: y = ⎨ x x < 0
⎪0 x = 0
⎩
1
x2
AIIF
Non–Linear Functions
Assessment
Page 197
Name: ___________________________
Non–Linear Functions Assessment
Formulas and Definitions
Quadratic Equation Standard Form:
y = ax 2 + bx + c
Quadratic Equation General Form:
ax 2 + bx + c = 0
Piece–Wise Function: For example,
for x <0
⎧5
y=⎨
⎩2x+5 for x ≥ 0
General Equation of a Circle with Center at (0, 0)
−b ± b 2 − 4 ac
2a
Break Even Point – The point where the revenue,
R(x), equals the cost, C(x). Symbolically, R(x) = C(x).
Also where the profit function P(x) = 0.
−b
Vertex – x coordinate: x =
2a
⎛ −b ⎞
Vertex – y coordinate: y = f ⎜ ⎟
⎝ 2a ⎠
−b
Line of Symmetry: x =
2a
Exponential Growth: y = Cax, a > 1, C > 1
Exponential Decay: y = Ca–x, a > 1, C > 1
and Radius r: x 2 + y 2 = r 2
General Equation of a Circle with Center at (h, k)
Quadratic Formula: x =
2
2
and Radius r: ( x − h ) + ( y − k ) = r 2
Discriminant: b 2 − 4 ac
Power Function: y = ax n
k
x
Floor Function: y = ⎣⎢ x ⎦⎥ , int() function on the
Inverse Variation: y =
graphing calculator
Ceiling Function: y = ⎢⎡ x ⎥⎤
Absolute Value Function: y = x
Complete two of the Problems 1 through 4.
1.
Draw a graph of a quadratic function with a line of symmetry of x = –3 and x–intercepts of x = 2 and
x = –8.
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Page 198
2.
Determine the quadratic equation from the graph at the
right.
3.
Write a quadratic function with no real number roots
and show why it has no real number roots.
4.
The total costs for a company are given by
C ( x ) = 2000 + 40 x + x 2 . The total revenues are given by
R( x ) = 250 x .
a.
What does the 2000 in C(x) represent?
b.
What does the 250 in R(x) represent?
c.
Find the profit function (profit = revenue minus
cost).
d.
Find the break-even points (profit = 0).
e.
For what values of x, does the company make a profit?
f.
What is the maximum profit? How many units must be produced and sold to maximize profit?
Non–Linear Functions
Assessment
AIIF
Page 199
Complete two of the Problems 5 through 8. Use the coordinate grid below for problems 5 and 6. Label your
graphs.
5.
Draw the graph of an odd function.
6.
Draw the graph of an even function that has a
maximum value of 4.
7.
Write an equation of a power function and then
rewrite the equation so that it is reflected about the
x–axis.
8.
Write an equation of a power–like function that has a
vertical translation of –3 units from the origin and a
horizontal translation of +4 units from the origin.
Complete two of the Problems 9 through 12. Use the coordinate grid below for problems 9 and 10.
9.
Sketch the graph of an exponential decaying
function.
10.
Draw the graph of an exponential growth function
with y–intercept of (0, 5). Explain the meaning of the
value of the y coordinate of the y–intercept.
11.
Explain the difference between a power function and an exponential function. Give an example of each
type of function.
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Page 200
12.
A breeder reactor converts stable uranium-238 into isotope plutonium239. The decay of this isotope is given by A(t ) = A0 e −0.00002876t , where
A(t) is the amount of isotope at time t, in years, and A0 is the initial
amount . If A0 = 1000 pounds, how much will be left after a human
lifetime (use t = 70 years)? Round your answer to the nearest
hundredth of a pound. Use 2.718 for e.
Complete two of the Problems 13 through 15.
13.
The amount of time, t, it takes to complete a job varies inversely with the amount of people, p, available
to complete the job. If t = 5 when p = 4, what is the value of t when p = 20? What is the value for the
constant of variation k?
14.
What restrictions, if any, must be made on an equation of a circle so that the equation of a circle
represents a function?
15.
Solve the circle equation, ( x − 4)2 + ( y + 2)2 = 25 for y. State the radius and the center of the circle.
Complete two of the Problems 16 through 19.
16.
Write the equation of an absolute value function that
has been horizontally translated –3 units from the
origin and vertically translated +2 units from the
origin.
17.
Draw the graph of the function in Problem 16.
AIIF
Non–Linear Functions
Assessment
Page 201
18.
Write a piece–wise function that is linear for x ≤ 0 and varies inversely for x > 0.
19.
You have taken a part–time job after school to save
for a college education. Your starting salary is $10.00
per hour. The store pays its employees for each
fifteen minute interval they work.
20.
a.
Write a step function equation based on the
information in the exercise.
b.
Create a table of values for an appropriate x
interval.
c.
Graph your step function equation.
For the matrix below, for each function and corresponding characteristic that relates to the function put
an A if it is always true, N if it is never true, or S if it is sometimes true. For the S characteristic, give an
example where it is sometimes true.
Has
Constant
Exponents
Quadratic
Function
Power
Function
Exponential
Function
Inverse
Function
Absolute
Value
Function
Piece–Wise
Function
Circle
Function
Step
Function
Has
Variable
Exponents
Can be
Horizontally
and/or
Vertically
Translated
N/A
Has a
Maximum
or
Minimum
Domain or
Range
Must be
Restricted
Has
Symmetry
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Page 202
References & Resources
The authors and contributors of Algebra II Foundations gratefully acknowledges the following resources:
Donovan, Suzanne M.; Bransford, John D. How Students Learn Mathematics in the Classroom. Washington, DC: The
National Academies Press. 2005.
Driscoll, Mark. Fostering Algebraic Thinking: A Guide for Teachers Grades 6-10. Portsmouth, NH: Heinemann, 1999.
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1983.
Harmin, Merrill. Inspiring Active Learning: A Handbook for Teachers. Alexandria, VA: Association for Supervision and
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Hoffman, Mark S, ed. The World Almanac and Book of Facts 1992. New York, NY: World Almanac. 1992.
Kagan, Spencer. Cooperative Learning. San Clemente, CA: Resources for Teachers. 1994.
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McIntosh, Alistair, Barbara Reys, and Robert Reys. Number Sense: Simple Effective Number Sense Experiences. Parsippany,
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McTighe, Jay; Wiggins, Grant. Understand by Design. Alexandria, VA: Association for Supervision and Curriculum
Development. 2004.
Marzano, Robert J. Building Background Knowledge for Academic Achievement. Alexandria, VA: Association for
Supervision and Curriculum Development. 2004.
Marzano, Robert J.; Pickering, Debra J.; Jane E. Pollock. Classroom Instruction that Works. Alexandria, VA:
Association for Supervision and Curriculum Development. 2001.
National Research Council. Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Academy Press.
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Ogle, D.M. (1986, February). “K-W-L: A Teaching Model That Develops Active Reading of Expository Text.” The Reading
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Van de Walle, Jon A. Elementary and Middle School Mathematics: Teaching Developmentally (4th Edition). New York: Addison
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Non–Linear Functions
Assessment
AIIF
Page 203
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