PreCalculus
Curriculum Timeline
GRADING PERIOD 1
Topic 1: Functions and Graphs
Topic 2: Polynomial, Power, and Rational Functions
Learning Goal A
Learning Goal B
Learning Goal C
Learning Goal D
Learning Goal E
Learning Goal F
Learning Goal A
Learning Goal B
Learning Goal C
Learning Goal D
No. of days: 40-45 (Continued in 2nd 9-Weeks)
No. of days: 25-30
GRADING PERIOD 2
Topic 2: Polynomial, Power, and Rational Functions
Topic 3: Exponential, Logarithmic, and Logistic Functions
Learning Goal A
Learning Goal B
Learning Goal C
Learning Goal D
Learning Goal A
Learning Goal B
Learning Goal C
Learning Goal D
No. of days: 25-30 (Continued in 3rd 9-Weeks)
No. of days: 40-45 (Continued from 1st 9-Weeks)
GRADING PERIOD 3
Topic 3: Exponential, Logarithmic, and Logistic Functions
Topic 4: Trigonometry and Trigonometric Functions
Learning Goal A
Learning Goal B
Learning Goal C
Learning Goal D
Learning Goal A
Learning Goal F
Learning Goal B
Learning Goal G
Learning Goal C
Learning Goal H
Learning Goal D
Learning Goal I
Learning Goal E
Learning Goal J
No. of days: 14-16 (Continued in 4th 9-Weeks)
No. of days: 25-30 (Continued from 2nd 9-Weeks)
GRADING PERIOD 4
Topic 4: Trigonometry and Trigonometric Functions
Topic 5: Noncartesian Representations
Learning Goal A
Learning Goal F
Learning Goal B
Learning Goal G
Learning Goal C
Learning Goal H
Learning Goal D
Learning Goal I
Learning Goal E
Learning Goal J
No. of days: 14-16 (Continued from 3rd 9-Weeks)
Learning Goal A
Learning Goal B
Learning Goal C
Learning Goal D
June 2005
No. of days: 10-15
PreCalculus
1st Nine-Weeks
Scope and Sequence
Topic 1: Functions and Graphs (25-30 days)
A) Identifies properties of functions by investigating intercepts, zeros, domain, range, horizontal
and vertical asymptotes, and local and global behavior and uses functions to model problems
B) Identifies the characteristics of the following families of functions: polynomials of degree
one, two and three, reciprocal, square root, exponential, logarithmic, sine, cosine, absolute
value, greatest integer and logistic
C) Performs operations with functions, including sum, difference, product, quotient, and
composition and transformations.
D) Represents the inverse of a function symbolically and graphically as a reflection about the
line y=x.
E) Identifies families of functions with graphs that have reflectional symmetry about the y-axis,
x-axis, or y=x.
F) Analyzes and interprets bivariate data to identify patterns, note trends, draw conclusions, and
make predictions.
Topic 2: Polynomial, Power, and Rational Functions (40 – 45 days)
(Continued in 2nd Nine-Weeks)
A) Determines the characteristics of the polynomial functions of any degree, general shape,
number of real and nonreal (real and nonreal), domain and range, and end behavior, and
finds real and nonreal zeros.
B) Identifies power functions and direct and inverse variation.
C) Describes and compares the characteristics of rational functions; e.g., general shape, number
of zeros (real and nonreal), domain and range, asymptotic behavior, and end behavior.
D) Analyzes and interprets bivariate data to identify patterns, note trends, draw conclusions, and
make predictions.
COLUMBUS PUBLIC SCHOOLS
MATHEMATICS CURRICULUM GUIDE
SUBJECT
PreCalculus
STATE STANDARDS 4 and 5
Patterns, Functions, and Algebra
Data Analysis and Probability
TIME RANGE
25-30 days
GRADING
PERIOD
1
MATHEMATICAL TOPIC 1
Functions and Graphs
A)
B)
C)
D)
E)
F)
CPS LEARNING GOALS
Identifies properties of functions by investigating intercepts, zeros, domain, range, horizontal
and vertical asymptotes, and local and global behavior and uses functions to model problems
Identifies the characteristics of the following families of functions: polynomials of degree
one, two and three, reciprocal, square root, exponential, logarithmic, sine, cosine, absolute
value, greatest integer and logistic.
Performs operations with functions, including sum, difference, product, quotient, and
composition and transformations.
Represents the inverse of a function symbolically and graphically as a reflection about the
line y = x.
Identifies families of functions with graphs that have reflectional symmetry about the y-axis,
x-axis, or y = x.
Analyzes and interprets bivariate data to identify patterns, note trends, draw conclusions, and
make predictions.
COURSE LEVEL INDICATORS
Course Level Indicators (i.e., How does a student demonstrate mastery?):
9 Identifies the intervals on which a function is increasing or decreasing. Math A:11-A:04
9 Uses interval notation to describe domain and range and the solution to inequalities.
Math MP:11/12-H
9 Identifies removable, jump, and infinite discontinuities. Math A:11-A:03
9 Identifies boundedness of a function and intervals on which a function is bounded.
Math A:11-A:03
9 Uses limit notation to describe asymptotic and end behaviors. Math A:12-A:07
9 Solves equations, inequalities, and systems of equations and inequalities graphically and
algebraically. Math A:11-A:03
9 Identifies points of discontinuity and the intervals over which a function is continuous.
Math A:11-A:03
9 Models real world data with functions. Math A:11-A:03, Math MP:11-D:11, and
Math D:11-A:04
9 Determines points of discontinuity and intervals on which a function is continuous.
Math A:11-A:03
9 Sketches graphs of basic functions and their transformations without technology.
Math A:11-A:03
9 Connects geometric transformations on the graph to changes of parameters in an equation.
Math A:11-A:03
9 Finds the composition of two or more functions. Math A:11-A:03
PreCalculus Standards 4 and 5
Functions and Graphs
Page 1 of 73
Columbus Public Schools 7/20/05
9
9
9
9
Writes a given function as the composition of simple functions. Math A:11-A:03
Determines the equation of an inverse relation. Math A:11-A:06
Uses the horizontal line test to determine if a relation is one-to-one. Math A:11-A:03
Explains the concept of even and odd functions and uses algebraic tests to determine
symmetry. Math A:11-A:05
Previous Level:
9 Describes the behavior of functions involving absolute value. Math A:11-A:05
9 Determines the domain and range of a function. Math A:09-E:01
9 Defines function and uses function notation. Math A:10-E:01
9 Explains the concept of inverse relationships and reflections about the line y=x.
Math A:11-A:06
9 Uses the vertical line test to determine if a relation is a function. Math A:10-B:01
9 Uses technology to find the Least Squares Regression Line, the regression coefficient, and
the correlation coefficient for bivariate data with a linear trend, and interpret each of these
statistics in the context of the problem situation. Math D:11-B:05
Next Level:
9 Analyzes functions by investigating rates of change. Math A:12-A:10
PreCalculus Standards 4 and 5
Functions and Graphs
Page 2 of 73
Columbus Public Schools 7/20/05
The description from the state, for the Patterns, Functions, and Algebra Standard says:
Students use patterns, relations, and functions to model, represent, and analyze problem
situations that involve variable quantities. Students analyze, model and solve problems using
various representations such as tables, graphs, and equations.
The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is:
A. Analyze functions by investigating rates of change, intercepts, zeros, asymptotes and local
and global behavior.
The description from the state, for the Data Analysis and Probability Standard says:
Students pose questions and collect, organize, represent, interpret, and analyze data to answer
those questions. Students develop and evaluate inferences, predictions, and arguments that are
abased on data.
The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is:
A. Create and analyze tabular and graphical displays of data using appropriate tools, including
spreadsheets and graphing calculators.
The description from the state, for the Mathematical Processes Standard says:
Students use mathematical processes and knowledge to solve problems. Students apply
problem-solving and decision-making techniques, and communicate mathematical ideas.
The grade-band benchmarks from the state, for this topic in the grade band 11 – 12 are:
D. Select and use various types of reasoning and methods of proof.
H. Use formal mathematical language and notation to represent ideas, to demonstrate
relationships within and among representation systems, and to formulate generalizations.
J. Apply mathematical modeling to workplace and consumer situations including problem
formulation, identification of a mathematical model, interpretation of solution within the
model, and validation to original problem situation.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 3 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs - A
Given the graph below, which is the correct description of the discontinuity?
A. The discontinuity is removable and is an infinite discontinuity.
B. The discontinuity is nonremovable and is an infinite discontinuity.
C. The discontinuity is nonremovable and is a jump discontinuity.
D. The discontinuity is removable and is a jump discontinuity.
Given y = (5 − x)3 + 3x − 10 , which statement is true?
A. The function is always decreasing
B. The function is always increasing.
C. The domain of the function is (-∞, 5].
D. The domain of the function is (-6.5, 5).
PreCalculus Standards 4 and 5
Functions and Graphs
Page 4 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –A
Answers/Rubrics
Low Complexity
Given the graph below, which is the correct description of the discontinuity?
A. The discontinuity is removable and is an infinite discontinuity.
B. The discontinuity is nonremovable and is an infinite discontinuity.
C. The discontinuity is nonremovable and is a jump discontinuity.
D. The discontinuity is removable and is a jump discontinuity.
Answer: C
Moderate Complexity
Given y = (5 − x)3 + 3x − 10 , which statement is true?
A. The function is always decreasing
B. The function is always increasing.
C. The domain of the function is (-∞, 5].
D. The domain of the function is (-6.5, 5).
Answer: C
PreCalculus Standards 4 and 5
Functions and Graphs
Page 5 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –A
Given the graph below, which statement correctly describes the behavior of the function?
A. lim f (x) = ∞
x→∞
B. lim f (x) = 0
x→−∞
C. lim f (x) = ∞
x→1
D. lim f (x) = 0
x→−1
x+2
contains a discontinuity. Use a graphing calculator to
x −1
sketch the graph and identify the type of discontinuity. Show algebraically how you verify the
discontinuity.
The graph of the function f ( x) =
PreCalculus Standards 4 and 5
Functions and Graphs
Page 6 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –A
Answers/Rubrics
High Complexity
Given the graph below, which statement correctly describes the behavior of the function?
A. lim f (x) = ∞
x→∞
B. lim f (x) = 0
x→−∞
C. lim f (x) = ∞
x→1
D. lim f (x) = 0
x→−1
Answer: B
Short Answer/Extended Response
x+2
contains a discontinuity. Use a graphing calculator to
x −1
sketch the graph and identify the type of discontinuity. Show algebraically how you verify the
discontinuity.
The graph of the function f ( x) =
This is an infinite discontinuity at x = 1. It occurs
because the denominator x - 1 = 0 at x = 1.
A 2 point response correctly identifies the infinite
discontinuity and its position.
A 1 point response identifies the infinite discontinuity.
A 0 point response show no mathematical
understanding.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 7 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Use a graphing calculator to graph f (x) =
Functions and Graphs –A
x+2
. Choose the statement which is false.
3x 2 − 5
A. The value of the function at x = 3 is 25.
B. lim f (x) = 0 .
x→−∞
C. The range is the set of real numbers.
D. lim f (x) = 0 .
x→∞
⎧ 3x 2 − 2, x ≤ 0
Use the function y = ⎨
Which statement is true?
−2x
+
1,
x
>
0
⎩
A. The value of the function at x = 3 is 25.
B. The relation is continuous.
C. The value of the function at x = -1 is -3.
D. The relation is a function.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 8 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –A
Answers/Rubrics
Low Complexity
Use a graphing calculator to graph f (x) =
x+2
. Choose the statement which is false.
3x 2 − 5
A. The value of the function at x= 3 is 25.
B. lim f (x) = 0
x→−∞
C. The range is the set of real numbers.
D. lim f (x) = 0
x→∞
Answer: A
Moderate Complexity
⎧ 3x 2 − 2, x ≤ 0
Use the function y = ⎨
Which statement is true?
⎩−2x + 1, x > 0
A. The value of the function at x = 3 is 25.
B. The relation is continuous.
C. The value of the function at x = -1 is -3.
D. The relation is a function.
Answer: D
PreCalculus Standards 4 and 5
Functions and Graphs
Page 9 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –A
The base of an isosceles triangle is half as long as the two equal sides. Which of the following
gives the area of the triangle (A) as a function of the length of the base (b).
A. A =
15b 2
4
B. A =
3b 2
2
C. A =
5b 2
4
D. A =
3b 2
2
2 x2 − 2
using a graphing calculator. Identify the x-intercepts,
x2 − 4
asymptotes, domain and range.
Graph the function f ( x) =
PreCalculus Standards 4 and 5
Functions and Graphs
Page 10 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –A
Answers/Rubrics
High Complexity
The base of an isosceles triangle is half as long as the two equal sides. Which of the following
gives the area of the triangle (A) as a function of the length of the base (b)?
15b 2
A. A =
4
B. A =
3b 2
4
5b 2
C. A =
4
D. A =
3b 2
2
Answer: A
Short Answer/Extended Response
2 x2 − 2
using a graphing calculator. Identify the x-intercepts,
x2 − 4
10
asymptotes, domain and range.
8
6
Answer:
Graph the function f ( x) =
4
2
-8 -7 -6 -5 -4 -3 -2 -1
-2
1 2 3 4 5 6 7 8
-4
-6
-8
-10
The zeros are 1 and -1. The vertical asymptotes are x = 2 and x = -2, the domain is
(−∞, −2) ∪ (−2, 2) ∪ (2, ∞ ) , and the range is (-∞, 2) ∪ (2,∞) .
A 2-point response correctly identifies the intercepts, asymptotes, domain and range.
A 1-point response correctly identifies 2 of the 4 required answers: the x-intercepts, the
asymptotes, the domain, and the range.
A 0-point response shows no mathematical understanding.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 11 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Which is the graph of y = sin x on (-2π, 2π)?
A.
B.
C.
Functions and Graphs –B
D.
Which function(s) have domain (-∞, ∞)?
A. y = cos x
B. y = ln x
C. y = 1/x
D. all of the above
E. none of the above
PreCalculus Standards 4 and 5
Functions and Graphs
Page 12 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –B
Answers/Rubrics
Low Complexity
Which is the graph of y= sin x on (-2π, 2π)?
A.
B.
C.
D.
Answer: B
Moderate Complexity
Which function(s) have domain (-∞, ∞)?
A. y = cos x
B. y = ln x
C. y = 1/x
D. all of the above
E. none of the above
Answer: A
PreCalculus Standards 4 and 5
Functions and Graphs
Page 13 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –B
Which letter represents a group of functions that are all bounded below?
A. f(x) = ln x, f(x) = x2, f ( x) = x
B. f(x) = sin x, f ( x) =
1
, f ( x) = x
1 + e− x
C. f(x) = sin x, f(x) = ln x
D. f(x) = cos x, f ( x) = e x , f ( x) =
1
1 + e− x
⎧− x if x ≤ 0
Sketch the graph of the piecewise function f ( x) = ⎨
. What basic function does
⎩ x if x ≥ 0
this represent? Why?
PreCalculus Standards 4 and 5
Functions and Graphs
Page 14 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Functions and Graphs –B
High Complexity
Which letter represents a group of functions that are all bounded below?
A. f(x ) = ln x, f(x) = x2, f ( x) = x
B. f(x) = sin x, f ( x) =
1
, f ( x) = x
1 + e− x
C. f(x) = sin x, f(x) = ln x
D. f ( x) = x , f(x) = cos x, f ( x) = e x , f ( x) =
1
1 + e− x
Answer: D
Short Answer/Extended Response
⎧− x if x ≤ 0
Sketch the graph of the piecewise function f ( x) = ⎨
. What basic function does
⎩ x if x ≥ 0
this represent? Why?
Answer:
The piecewise function is the same as the absolute
value function because the definition of absolute value
says for x ≥ 0, the absolute value of x= x and for x ≤ 0, the
absolute value of x = -x.
A 2-point response identifies the absolute value function and supports the answer.
A 1-point response identifies the absolute value function without support.
A 0-point response shows no mathematical understanding.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 15 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –C
Which represents (f + g) (x) if f(x) = x2 and g ( x) = x + 1 for x ≥ 0?
A. (f + g) (x) =
x2 + 1
B. (f + g) (x) =
x2 + x + 1
C. (f + g) (x) = x 2 + x + 1
D. (f + g) (x) = x + 1 + x
What is the domain of
f ( x)
if f(x) = x2 and g ( x) = x + 1 ?
g ( x)
A. x ≠ 0
B. All real numbers
C. x ≠ 1
D. x ≠ -1
PreCalculus Standards 4 and 5
Functions and Graphs
Page 16 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –C
Answers/Rubrics
Low Complexity
Which represents (f + g) (x) if f(x) = x2 and , g ( x) = x + 1 for x ≥ 0?
A. (f + g) (x) =
x2 + 1
B. (f + g) (x) =
x2 + x + 1
C. (f + g) (x) = x 2 + x + 1
D. (f + g) (x) = x + 1 + x
Answer: B
Moderate Complexity
What is the domain of
f ( x)
if f(x) = x2 and g ( x) = x + 1 ?
g ( x)
A. x ≠ 0
B. All real numbers
C. x ≠ 1
D. x ≠ -1
Answer: D
PreCalculus Standards 4 and 5
Functions and Graphs
Page 17 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –C
Which represents (f ◦ g) (x) if f(x) = x2 and g ( x) = x + 1 ?
A. (f ◦ g) (x) =
x2 + 1
B. (f ◦ g) (x) = x + 1
C. (f ◦ g) (x) =
x2 + x + 1
D. (f ◦ g) (x) =
x2 + x
Given f(x) = x2 - 1 and g ( x) = x , find (f ◦ g) (x) and (g ◦ f ) (x) and the domain of each.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 18 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –C
Answers/Rubrics
High Complexity
Which represents (f ◦ g) (x) if f(x) = x2 and g ( x) = x + 1 ?
A. (f ◦ g) (x) =
x2 + 1
B. (f ◦ g) (x) = x + 1
C. (f ◦ g) (x) =
x2 + x + 1
D. (f ◦ g) (x) =
x2 + x
Answer: B
Short Answer/Extended Response
Given f(x) = x2 - 1 and g ( x) = x , find (f ◦ g) (x) and (g ◦ f ) (x) and the domain of each.
Answer: (f ◦ g) (x )= x-1 and the domain is [0, ∞)
(g ◦ f) (x )=
x 2 − 1 and the domain is (-∞,-1) ∪ (1, ∞).
A 2 point response correctly identifies the compositions and their domains.
A 1-point response correctly identifies the compositions.
A 0- point response shows no mathematical understanding of the topic.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 19 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –D
Given the graph below. which is the graph of the inverse?
10
8
6
4
2
-10 -8 -6
-4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
A.
B.
C.
D.
10
10
10
10
8
8
8
8
6
6
6
6
4
4
4
4
2
2
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
-10 -8 -6
-4 -2
2
4
6
8
10
-10 -8 -6 -4 -2
-2
-4
-6
6
8 10
-10 -8 -6 -4 -2
2
-2
-4
-4
-6
-6
-10
-10
4
-2
-8
-8
2
2
4
6
8 10
-8
-10
Which is the inverse of f (x) = x 3 − 5 ?
A. f −1 (x) = 3 x − 5
B. f −1 (x) = 3 x − 5
C. f −1 (x) = 3 x + 5
D. f −1 (x) = 3 x + 5
PreCalculus Standards 4 and 5
Functions and Graphs
Page 20 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –D
Answers/Rubrics
Low Complexity
Given the graph below, which is the graph of the inverse?
10
8
6
4
2
-10 -8 -6
-4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
A.
B.
C.
D.
10
10
10
10
8
8
8
8
6
6
6
6
4
4
4
4
2
2
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
-10 -8 -6
-4 -2
2
4
6
8
10
-10 -8 -6 -4 -2
-2
-4
-6
2
6
8 10
2
-2
-4
-4
-6
-6
-10
-10
4
-2
-8
-8
2
-10 -8 -6 -4 -2
4
6
8 10
-8
-10
Answer: B
Moderate Complexity
Which is the inverse of f (x) = x 3 − 5 ?
A. f −1 (x) = 3 x − 5
B. f −1 (x) = 3 x − 5
C. f −1 (x) = 3 x + 5
D. f −1 (x) = 3 x + 5
Answer: D
PreCalculus Standards 4 and 5
Functions and Graphs
Page 21 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –D
Which function has an inverse that is a function?
A. f(x) = x3
B. f(x ) = ln x
C. f(x) = x
D. All of the above.
E. None of the above.
Give the equation of a function that is one-to-one and a function that is not one-to-one. Explain
your choices both algebraically and graphically.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 22 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –D
Answers/Rubrics
High Complexity
Which function has an inverse that is a function?
A. f(x) = x3
B. f(x) = ln x
C. f(x) = x
D. All of the above.
E. None of the above.
Answer: D
Short Answer/Extended Response
Give the equation of a function that is one-to-one and a function that is not one-to-one. Explain
your choices both algebraically and graphically.
Sample Answer: y = x 3 is one-to-one. When you interchange x and y and solve for y,
there is only one solution. The graph passes both the vertical and horizontal line tests.
y = x 2 is not one-to-one. When you interchange x and y and solve for y, there are two
solutions. The graph fails the horizontal line test.
A 2-point solution includes both a one-to-one function and a function which is not one-toone and supports the answer both graphically and algebraically.
A 1-point solution includes both a one-to-one function and a function which is not one-toone and supports the answer EITHER graphically OR algebraically.
A 0-point solution demonstrates no mathematical understanding.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 23 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –D
Given the graphs below, what symmetries are exhibited?
I
II
III
A. I: x-axis, II: y-axis, III: origin
B. I: y-axis, II: x-axis, III: origin
C. I: y-axis, II: origin, III: x-axis
D. I: origin, II: y-axis, III: x-axis
A graph of a relation which is symmetric about the x-axis contains the points (2, 3), (-5, 1), and
(-4, -6). Which other points must also be on the graph?
A. (2,-3), (5,1), and (-4, 6)
B. (2, -3), (-5, -1), and (-4, 6)
C. (-2, 3), (5,1), and (4, -6)
D. (-2, 3), (-5, -1), and (4, -6)
PreCalculus Standards 4 and 5
Functions and Graphs
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58PRACTICE ASSESSMENT ITEMS
Functions and Graphs –D
Answers/Rubrics
Low Complexity
Given the graphs below, what symmetries are exhibited?
I
II
III
A. I: x-axis, II: y-axis, III: origin
B. I: y-axis, II: x-axis, III: origin
C. I: y-axis, II: origin, III: x-axis
D. I: origin, II: y-axis, III: x-axis
Answer: C
Moderate Complexity
A graph of a relation which is symmetric about the x-axis contains the points (2, 3), (-5, 1), and
(-4, -6). Which other points must also be on the graph?
A. (2, -3), (5, 1), and (-4, 6)
B. (2, -3), (-5, -1), and (-4, 6)
C. (-2, 3), (5, 1), and (4, -6)
D. (-2, 3), (-5, -1), and (4, -6)
Answer: B
PreCalculus Standards 4 and 5
Functions and Graphs
Page 25 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –D
Given a function f such that f (− x) = − f (x) , which statement is true?
A. The function is symmetric about the x-axis.
B. The function is symmetric about the y-axis.
C. The function is symmetric about the origin.
D. The function does not necessarily exhibit symmetry.
Use your calculator to graph the function f (x) =
the symmetry algebraically.
PreCalculus Standards 4 and 5
Functions and Graphs
x2 − 9
. Describe the symmetry and verify
x2 − 4
Page 26 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –E
Answers/Rubrics
High Complexity
Given a function f such that f (− x) = − f (x) , which statement is true?
A. The function is symmetric about the x-axis.
B. The function is symmetric about the y-axis.
C. The function is symmetric about the origin.
D. The function does not necessarily exhibit symmetry.
Answer: C
Short Answer/Extended Response
Use your calculator to graph the function f (x) =
the symmetry algebraically.
x2 − 9
. Describe the symmetry and verify
x2 − 4
10
8
(− x ) − 9
f (− x ) =
( − x )2 − 4
2
6
4
x2 − 9
= 2
x −4
= f ( x)
Because f ( x ) = f ( − x ) , the function is symmetric about
the y-axis.
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
A 2-point response includes a correct graph, identifies the function as symmetric about
the y-axis, and provides algebraic support.
A 1-point response includes a correct graph and identifies the function as symmetric
about the y-axis.
A 0-point response demonstrates no mathematical understanding.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 27 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –E
Given the scatterplot below, which of the following types of regression is likely to give the most
accurate model?
A. exponential
B. sinusoidal
C. quadratic
D. quartic
The table shows the population of a certain city in various years.
Year
1981 1985 1989 1993 1997
Population (hundreds of thousands) 3.2
4.1
5.7
9.6
14.1
Using x as the number of years which have elapsed since 1980 and y as the population of the city
in hundreds of thousands, which exponential function models the data?
A. y = 1.8506(0.51209)x
B. y = 2.0127(1.4647) x
C. y = 1.7025(0.53750)x
D. y = 2.6797(1.10012) x
PreCalculus Standards 4 and 5
Functions and Graphs
Page 28 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –E
Answers/Rubrics
Low Complexity
Given the scatterplot below, which of the following types of regression is likely to give the most
accurate model?
A. exponential
B. sinusoidal
C. quadratic
D. quartic
Answer: B
Moderate Complexity
The table shows the population of a certain city in various years.
Year
1981 1985 1989 1993 1997
Population (hundreds of thousands) 3.2
4.1
5.7
9.6
14.1
Using x as the number of years which have elapsed since 1980 and y as the population of the city
in hundreds of thousands, which exponential function models the data?
A. y = 1.8506(0.51209)x
B. y = 2.0127(1.4647) x
C. y = 1.7025(0.53750)x
D. y = 2.6797(1.10012) x
Answer: D
PreCalculus Standards 4 and 5
Functions and Graphs
Page 29 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –E
The table below shows several examples of saturated vapor pressure and the associated relative
humidity.
Saturated Vapor Pressure (in millibars) 12.26 17.65 25.01 34.94 48.12 65.43
Relative Humidity %
100
69.5 49
35.1 25.5 18.7
Find an equation that models the data and use that equation to find the relative humidity if the
saturated vapor pressure is 41.3 millibars.
A. 27.6%
B. 29.7%
C. 31.6%
D. 33.7%
The table below shows the U.S. per capita income for the years 1990-2000.
Year
Amount (in $)
Year
Amount (in $)
1990
19,614
1996
24,660
1991
20,126
1997
25,876
1992
21,105
1998
27,317
1999
21,736
1999
28,534
1994
22,593
2000
30,069
1995
23,571
????
45,000
Using the year 1990 as t = 0, make a scatterplot of the data and find the most appropriate
regression model using a graphing calculator. Sketch the scatterplot and the graph of the
regression equation. Use the model to predict the year in which the per capita income will
exceed $45,000.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 30 of 73
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Functions and Graphs –E
Answers/Rubrics
High Complexity
The table below shows several examples of saturated vapor pressure and the associated relative
humidity.
Saturated Vapor Pressure (in millibars) 12.26 17.65 25.01 34.94 48.12 65.43
Relative Humidity %
100
69.5 49
35.1 25.5 18.7
Find an equation that models the data and use that equation to find the relative humidity if the
saturated vapor pressure is 41.3 millibars.
A. 27.6%
B. 29.7%
C. 31.6%
D. 33.7%
Answer: B
Short Answer/Extended Response
A 2-point response includes a correct, labeled
scatterplot with the graph of regression equation,
the regression equation, and a solution (either
graphical or algebraic) and an answer in terms of
the year.
Income ($)
The table below shows the U.S. per capita income for the years 1990-2000.
Year
1990
1991
1992
1993
1994
1995
Amount (in $) 19,614 20,126
21,105
21,736
22,593
23,571
Year
1996
1997
1998
1999
2000
????
Amount (in $) 24,660 25,876
27,317
28,534
30,069
45,000
Using the year 1990 as t =0, make a scatterplot of the data and find the most appropriate
regression model using a graphing calculator. Sketch the scatterplot and the graph of the
regression equation. Use the model to predict the year in which the per capita income will
exceed $45,000.
50000
Answer: The regression equation is
)
$
y = 49.1 x 2 + 553.5 x + 19623.3 . The model
(
(15.5007, 40000.)
40000
predicts that the per capita income will exceed
e
m
$45,000 in 2005.
o
c
n
I
30000
20000
5
10
Year
15
20
A 1-point response includes a correct scatterplot and graph and solution but does not
express the answer in terms of the year.
A 0-point response demonstrates no mathematical understanding.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 31 of 73
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Teacher Introduction
Functions and Graphs
This topic provides the foundation for the rest of the course. Many of the learning goals were
introduced in Algebra II at a basic level. In this topic, students are required to draw upon their
previous work and apply it to new situations. Students often encounter difficulty with this. In
previous courses, the topics are fairly narrowly focused, and students do not need to draw upon
concepts from outside that topic. In PreCalculus, it is essential that they bring their previous
knowledge to bear on this general study of functions. In addition, they need to be prepared to
apply the concepts learned in this topic to the rest of the PreCalculus course as they study
families of functions in greater depth. This course consists mostly of the study of functions. The
learning goals in this topic are essential to the remainder of the course and cannot be rushed.
This guide has been created to be used in conjunction with the text, and pages that are indicated
in the resources are essential for the implementation of the curriculum.
The strategies used in this topic involve several different learning goals. They are not intended
to be completed in one day or even on consecutive days. It is essential that the class come
together to discuss the different parts frequently.
The grouping of the subjects in the learning goals does not necessarily indicate the order in
which they should be taught, and the nature of the topic requires that the learning goals be
integrated. The pacing guide and correlations demonstrate this. This topic will probably require
four to five weeks.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 32 of 73
Columbus Public Schools 7/20/05
TEACHING STRATEGIES/ACTIVITIES
Vocabulary: mathematical model, domain, range, function, removable discontinuity,
jump discontinuity, infinite discontinuity, continuity, increasing, decreasing, constant,
lower bound, upper bound, boundedness, local extrema, absolute extrema, odd function,
even function, asymptote, identity function, squaring function, cubing function,
reciprocal function, square root function, exponential function, natural logarithm
function, sine function, cosine function, absolute value function, greatest function,
logistic function, symmetry, piecewise function, composition, inverse, relation, implicit,
inverse, transformation, translation, rigid transformation, reflection, stretch, shrink,
regression, correlation coefficient, quadratic, end behavior, zeros.
Core:
Learning Goal A: Identifies properties of functions by investigating intercepts, zeros, domain,
range, horizontal and vertical asymptotes, and local and global behavior and uses functions to
model problems.
1. Emphasize that the correct use of vocabulary is essential. Students must understand how
zeros, x-intercepts, and factors are related and use the correct word to describe each. With
rational functions, look at specific examples and then generalize.
2. Do the activity “The Bathtub” (included in this Curriculum Guide).
3. Do the activity “Piecewise Functions Step by Step” (included in this Curriculum Guide).
Learning Goal B: Identifies the characteristics of the following families of functions:
polynomials of degree one, two and three, reciprocal, square root, exponential, logarithmic,
sine, cosine, absolute value, greatest integer and logistic
1. Begin the activity, "Stacks of Cups" (included in this Curriculum Guide). This activity
includes many of the ideas included in this topic including the idea of slope as a rate of
change, and the greatest integer function. It also introduces the idea of inverse functions that
will be studied and points out the difference between the domain and the range of a function
and the situation that it models. This activity will be used throughout Topic One.
Learning Goal C: Performs operations with functions, including sum, difference, product,
quotient, and composition and transformations.
1. Do the activity “Transformations” (included in this Curriculum Guide.). This activity
reinforces the basic functions
Learning Goal D Represents the inverse of a function symbolically and graphically a reflection
about the line y = x.
1. Introduce inverse functions by using the Teacher Notes (included in this Curriculum Guide.)
Emphasize that the inverse of a function may not be a function. Students should investigate
functions that are and are not one-to-one and restricting the domain and range of functions.
Complete “Stacks of Cups” (included in this Curriculum Guide and started in Learning Goals
B).
PreCalculus Standards 4 and 5
Functions and Graphs
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Learning Goal E: Identifies families of functions with graphs that have reflection symmetry
about the y-axis, x-axis, or y = x.
1. Introduce symmetry by using the activity “Introduction to Symmetry” (included in this
Curriculum Guide). This is not actually an activity, but is really a graphic organizer for their
notes.
Learning Goal F: Analyzes and interprets bivariate data to identify patterns, note trends, draw
conclusions, and make predictions.
1. Do the activity “The Mile Run” (included in this Curriculum Guide.). This activity should
take several days while the class is continuing to study from the textbook. Parts of this
should be worked in groups in the classroom and parts should be assigned to be completed as
homework.
Reteach:
1. Review linear equations, x-intercepts, y-intercepts.
2. Review quadratic equations, factoring, use of the quadratic formula.
3. Complete the activity “Concepts in Graphical Analysis” (included in this Curriculum Guide.)
PreCalculus Standards 4 and 5
Functions and Graphs
Page 34 of 73
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RESOURCES
Learning Goal A:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 81-100, 131-141
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 13-14, 19-21
Learning Goal B:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 101-111
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp.15-16
Learning Goal C:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 122-128
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 17-18
Learning Goal D:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 93-95
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp.13-14
Learning Goal E:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 63-80. 152-156
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp.11-12, 21-22
PreCalculus Standards 4 and 5
Functions and Graphs
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Introduction to Inverse
Teacher Notes –D
The graphing calculator helps students to understand the idea of inverse, the difference between
an inverse relation and function, and one-to-one functions.
Start with the graphs of y= x3 in Y1 (with the “heavy” style) and y = x in Y2. From the home
screen, go to the DRAW menu and choose #8: DrawInv. (Locate Y1 by VARS, YVARS, 1:
Function, 1:Y1.) The resulting graph is a drawing, not a graph. You cannot trace on it or access
the points on the table. However, you can discuss the idea of reversing ordered pairs and the
reflection about the line y = x.
Repeat with a graph of y = x2in (with the “heavy” style) and the line y = x. It is obvious that by
just reversing the ordered pairs, the result is not necessarily a function, i.e. not all functions are
one-to-one. This points out the necessity for restricting the domain of the function in order to
create an inverse function. To restrict the domain of y = x2 to values of x that are greater than or
equal to zero, use the TEST (2nd MATH) menu as shown below in Y4. See the Graphing
Calculator Resource Manual (included in this Curriculum Guide) for a further discussion of the
use of the TEST menu.
By restricting the
domain and drawing
the inverse, the result
is one-to-one.
Students should
discuss how to find the
function that represents
the inverse (Y2) and
then test their
conjectures. You can
then find the other
branches, using Y3 and
Y4. Introducing the
line y = x may make
this more obvious.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 36 of 73
Columbus Public Schools 7/20/05
The Bathtub
Functions and Graphs –A
Name
Change in water level (in inches)
Below is a graph of the change in water level in a bathtub. At time t = 0, there is some unknown
amount of water in the tub. The graph tracks how the water level changes over time.
1. Identify time intervals over which the water level is
increasing.
a.
When does the bather enter the bathtub? How can
you tell?
b.
Other than that, when is the water level increasing
the fastest?
Time (in minutes)
2. Identify time intervals over which the water level is decreasing.
a.
When does the bather exit the bathtub? How can you tell?
b.
Other than that, when is the water level decreasing the fastest?
3. Where are the x-intercepts of the graph? What do they mean in terms of the problem
situation?
4. What is happening to the water depth as the time nears 16 minutes? How far below the
initial water level is the bottom of the tub?
5. What was the initial water level? What was the greatest depth of the water in the tub?
PreCalculus Standards 4 and 5
Functions and Graphs
Page 37 of 73
Columbus Public Schools 7/20/05
The Bathtub
Answer Key
Functions and Graphs –A
Change in water level (in inches)
Below is a graph of the change in water level in a bathtub. At time t = 0, there is some unknown
amount of water in the tub. The graph tracks how the water level changes over time.
1. Identify time intervals over which the water level is
increasing. (0, 4)
a. When does the bather enter the bathtub?
How can you tell? After 4 minutes; the
water level increases instantaneously
Time (in minutes)
b. Other than that, when is the water level
increasing the fastest? Between 2 and 4
minutes.
2 Identify time intervals over which the water level is decreasing.
Between 10 and 16 minutes. (10, 16)
a. When does the bather exit the bathtub? How can you tell?
After 12 minutes; the water level decreases instantaneously
b. Other than that, when is the water level decreasing the fastest?
Between 12 and 13 minutes
3. Where are the x-intercepts of the graph? What do they mean in terms of the problem
situation.?
At 0 and 12 minutes. This is when the water level is the same as the initial value.
4. What is happening to the water depth as the time nears 16 minutes? How far below the
initial water level is the bottom of the tub?
The level is getting close to zero. About 3 inches.
5. What was the initial water level? What was the greatest depth of the water in the tub?
About 3 inches. About 8 inches.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 38 of 73
Columbus Public Schools 7/20/05
Piecewise Functions Step by Step
Name
Functions and Graphs –A
To graph the piecewise function:
1 2x
if x -2
if - 2 x 1
f(x) 3
2
x 4 if x 1
1. Show the breaking points on the number-line below, and indicate for which x we will be
graphing which function:
-10
-8
-6
-4
-2
0
2
4
6
8
10
2. Graph the first piece of the function. Then, cross out the part you’re not using.
3. Graph the second piece of the function. Then, cross out the part you’re not using.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 39 of 73
Columbus Public Schools 7/20/05
4. Graph the third piece of the function. Then, cross out the part you’re not using.
Functions and Graphs –A
5. Now put all three pieces together in the following grid.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 40 of 73
Columbus Public Schools 7/20/05
Functions and Graphs –A
if x < -2
⎧1 − 2 x
⎪
To graph the piecewise function f ( x) = ⎨3
if - 2 ≤ x ≤ 1 on your graphing calculator, you
2
⎪− x + 4 if x > 1
⎩
must tell it which part to cross out. You do this with the test menu.
To enter access the inequality symbols in the TEST menu (2nd MATH.)
The inequality symbols are Boolean operators that are assigned the
value 0 when the expression is false and the value 1 when the expression
is true. The / leaves the equation unchanged when for the correct interval
and divides by zero when the expression is false.
Try these with and without your calculator.
⎧ 3 if x ≤ −1
6. f (x) = ⎨
⎩−2 if x > −1
⎧−2x if x < −1
⎪
7. f (x) = ⎨ x 2 if − 1 ≤ x < 1
⎪−2 if x ≥ 1
⎩
⎧ x − 3 if x ≤ −2
⎪
if − 2 < x < 1
8. f (x) = ⎨−x 2
⎪−x + 4 if x ≥ 1
⎩
PreCalculus Standards 4 and 5
Functions and Graphs
Page 41 of 73
Columbus Public Schools 7/20/05
Functions and Graphs –A
Piecewise Functions Step by Step
Answer Key
To graph the piecewise function:
1 2x
if x -2
f ( x)
3
if - 2 x 1
2
x 4 if x 1
1. Show the breaking points on the number-line below, and indicate for which x we will be
graphing which function:
1-2x
-10
-8
-6
3
-4
-2
x2 4
0
2
4
6
8
10
2. Graph the first piece of the function. Then, cross out the part you’re not using.
10
8
6
4
2
-10 -8 -6 -4 -2
-2
2
4
6
8 10
-4
-6
-8
-10
3. Graph the second piece of the function. Then, cross out the part you’re not using.
10
8
6
4
2
-10 -8 -6 -4 -2
-2
2
4
6
8 10
-4
-6
-8
-10
PreCalculus Standards 4 and 5
Functions and Graphs
Page 42 of 73
Columbus Public Schools 7/20/05
Functions and Graphs –A
4. Graph the third piece of the function. Then, cross out the part you’re not using.
10
8
6
4
2
-10 -8 -6 -4 -2
2
4
6
8
10
-2
-4
-6
-8
-10
5. Now put all three pieces together in the following grid.
10
8
6
4
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
PreCalculus Standards 4 and 5
Functions and Graphs
Page 43 of 73
Columbus Public Schools 7/20/05
Functions and Graphs –A
if x < -2
⎧1 − 2 x
⎪
To graph the piecewise function f ( x) = ⎨3
if - 2 ≤ x ≤ 1 on your graphing calculator, you
⎪− x 2 + 4 if x > 1
⎩
must tell it which part to cross out. You do this with the test
menu. To enter access the inequality symbols in the TEST
menu (2nd MATH.) The inequality symbols are Boolean
operators that are assigned the value 0 when the expression is
false and the value 1 when the expression is true. The / leaves
the equation unchanged when for the correct interval and
divides by zero when the expression is false.
10
8
Try these with and without your calculator.
⎧ 3 if x ≤ −1
6. f (x) = ⎨
⎩−2 if x > −1
6
4
2
-10 -8 -6 -4 -2
2
4
6
8 10
2
4
6
8 10
-2
-4
-6
-8
-10
10
8
6
⎧−2x if x < −1
⎪
7. f (x) = ⎨ x 2 if − 1 ≤ x < 1
⎪−2 if x ≥ 1
⎩
4
2
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
10
8
6
if x ≤ −2 ⎫
⎧ x −3
⎪
⎪
8. f ( x) = ⎨ − x 2 if − 2 < x < 1⎬
⎪− x + 4
if x ≥ 1 ⎪⎭
⎩
4
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
PreCalculus Standards 4 and 5
Functions and Graphs
Page 44 of 73
Columbus Public Schools 7/20/05
Stack of Cups
Functions and Graphs –B
Name
In this activity, you will investigate the relationship between the number of cups and
the height of the stack.
1.
Using a sample of cups, all of the same size, complete the
following chart. Use the data you collect to look for patterns that
might help you determine the relationship between the height of
the stack and the number of cups in that stack.
Number
of cups
Height of
stack (cm)
2. Make a scatterplot of the data, with the number of cups on the x-axis and the height of the
stack on the y-axis.
and 25cups: _
3. Use your graph to predict the height of a stack of 16 cups:
Circle the points on your graph that you used for your predictions.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 45 of 73
.
Columbus Public Schools 7/20/05
Functions and Graphs –B
4. Define a function, f, such that f(n) gives the height of a stack, h, in terms of the number of
cups in that stack, n. When you have your result, call in your teacher to verify your results.
Why is the y-intercept of your linear function NOT the same as the height of one cup?
What are the domain and range of the function that you wrote? How must you restrict these
to be the same as the domain and range of the function defined by the set of points (n,h)?
5. Sketch the graph of f on the grid with your scatterplot.
Use this information for #6-9. Another team used a different design of cup and found that the
equation that modeled the height in centimeters as a function of the number of cups to be
S(n)=.5n+12.5.
6. For this team:
a) What is the height of one cup?
b) What is change in height per cup?
c) If you increased the stack by 2 cups, how much would the height of the stack increase?
d) If you increased the stack by 20 cups, how much would the height of the stack increase?
e) In general, if you add k cups to an existing stack, how much will the height of the stack
increase?
7.
Suppose that another student claims that doubling the number of cups in the stack doubles
the height of the stack. Explain why this statement is incorrect. Support your argument with
examples.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 46 of 73
Columbus Public Schools 7/20/05
Functions and Graphs –B
8.
a) Using the cups from #6, if you increased the height of the carton by 5 cm, how many
more cups could you fit in?
b) If you increased the height of the carton by 6.4 cm, how many more cups could you fit in
the carton? Remember that you should not have a "part" of a cup.
c) In general, if the height of the carton were increased by d centimeters, how many more
cups could you fit in the carton?
9. How many cups could you fit in a carton of height 36 cm? 50 cm?
10. The function S(n)=.5n+12.5 expresses the height of a stack of cups in terms of the number of
cups in the stack. Now write a function g(h) that expresses the number of cups in a stack in
terms of the height, h, of the stack.
11. The slope of a line represents a rate of change.
What is the slope of S?
In the language of rate of change, this means that the height increases .5 cm for each increase
of 1 cup.
What is the slope of g? Express this as a rate of change.
S and g are inverses; that is, their ordered pairs are the reverse of each other. What is the
relationship of their slopes?
Use the concept of rate of change to explain why this is so.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 47 of 73
Columbus Public Schools 7/20/05
Stack of Cups
Answer Key
Functions and Graphs –B
In this activity, you will investigate the relationship between the number of cups and
the height of the stack.
1. Using a sample of cups, all of the same size, complete the
following chart. Use the data you collect to look for patterns that
might help you determine the relationship between the height of
the stack and the number of cups in that stack.
Number
of cups
Height of
stack (cm)
Answer will vary.
2. Make a scatterplot of the data, with the number of cups on the xaxis and the height of the stack on the y-axis.
Answer will vary.
and 25cups: __
3. Use your graph to predict the height of a stack of 16 cups:
Circle the points on your graph that you used for your predictions.
.
Answer will vary.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 48 of 73
Columbus Public Schools 7/20/05
Functions and Graphs –B
4. Define a function, f, such that f(n) gives the height of a stack, h, in terms of the number of
cups in that stack, n. When you have your result, call in your teacher to verify your results.
Why is the y-intercept of your linear function NOT the same as the height of one cup?
Answer will vary.
What are the domain and range of the function that you wrote? How must you restrict these
to be the same as the domain and range of the function defined by the set of points (n,h)?
Answer will vary.
5. Sketch the graph of f on the grid with your statplot.
Use this information for #6-9: Another team used a different design of cup and found that the
equation that modeled the height in centimeters as a function of the number of cups to be
S(n)=.5n+12.5.
6.
a) What is the height of one cup? 13 cm
b) What is change in height per cup? .5 cm
c) If you increased the stack by 2 cups, how much would the height of the stack increase?
1 cm
d) If you increased the stack by 20 cups, how much would the height of the stack increase?
10 cm
e) In general, if you add k cups to an existing stack, how much will the height of the stack
increase?
.5k
7.
Suppose that another student claims that doubling the number of cups in the stack doubles
the height of the stack. Explain why this statement is incorrect. Support your argument with
examples.
If we have 10 cups: s(10) =17.5
If we double this to 20 cups: s(20) = 22.5
22.5 is not double 17.5
PreCalculus Standards 4 and 5
Functions and Graphs
Page 49 of 73
Columbus Public Schools 7/20/05
8.
Functions and Graphs –B
d) Using the cups from #6, if you increased the height of the carton by 5 cm, how many
more cups could you fit in?
About 10 cups
e) If you increased the height of the carton by 6.4 cm, how many more cups could you fit in
the carton? Remember that you should not have a "part" of a cup.
6.4 = 12.8 which would be rounded to 12 cups.
.5
f) In general, if the height of the carton were increased by d centimeters, how many more
cups could you fit in the carton?
2d more cups
9. How many cups could you fit in a carton of height 36 cm?
50 cm? 47 cups, 75 cups
10. The function S(n)=.5n+12.5 expresses the height of a stack of cups in terms of the number of
cups in the stack. Now write a function g(h) that expresses the number of cups in a stack in
terms of the height, h, of the stack. g(h )= 2h - 25
11. The slope of a line represents a rate of change.
What is the slope of S? .5
In the language of rate of change, this means that the height increases .5 cm for each
increase of 1 cup.
What is the slope of g? Express this as a rate of change.
2; The number of cups increases by 2 for each 1 cm increase in height.
S and g are inverses; that is, their ordered pairs are the reverse of each other. What is the
relationship of their slopes? They are reciprocals
Use the concept of rate of change to explain why this is so. One is the change in cups over
change in cm and the other is just the reverse.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 50 of 73
Columbus Public Schools 7/20/05
Transformations
Functions and Graphs –C
Name
#1-2: Using the standard viewing window (ZOOM 6), graph the given parent function. Then
graph each of the given functions. (You should have only two graphs at a time.) Sketch the
graph and describe how the parent function could be moved to create the new function.
1. Parent function: f (x) = x 2
f (x) = x 2 + 2
f (x) = x 2 − 3
2. Parent function: f(x) = x
f (x) = x − 4
f (x) = x + 5
3. In general, if you have a parent function, y = f ( x ) , how will the graph of y = f (x) + c be
related to the parent function if c is positive? if c is negative?
PreCalculus Standards 4 and 5
Functions and Graphs
Page 51 of 73
Columbus Public Schools 7/20/05
Functions and Graphs –C
When you start with a parent function and change it, it is called a transformation. These
transformations are called vertical shifts. In #1, the transformations were vertical shifts of up 2
and down 3. In #2, the transformations were vertical shifts of down 4 and up 5.
4. Without using a calculator, sketch the graph of f (x) = x 3 . Then sketch the graph of
f (x) = x 3 − 3 . Check your answer by graphing on your calculator. What kind of transformation
is this?
5. Without using a calculator, sketch the graph of f (x) =
1
. Then sketch the graph of
x
1
+ 2 . Check your answer by graphing on your calculator. What kind of
x
transformation is this?
f (x) = f (x) =
#6-7: Using the standard viewing window (ZOOM 6), graph the given parent function. Then
graph each of the given functions. (You should have only two graphs at a time.) Sketch the
graph and describe how the parent function could be moved to create the new function.
6. Parent function: f (x) = x 3
f (x) = (x − 2)3
PreCalculus Standards 4 and 5
Functions and Graphs
f (x) = (x + 3)3
Page 52 of 73
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7. Parent function: f (x) = x
f (x) = (x − 1)
Functions and Graphs –C
f (x) = (x + 4)
8. In general, if you have a parent function, y = f (x) , how will the graph of y = f (x − b) be
related to the parent function if b is positive?
In general, if you have a parent function, y = f (x) , how will the graph of y = f (x − b) be related
to the parent function if b is negative?
Notice that if b is a positive number (like 3) the argument of the function will have a "-" e.g. x-3.
If b is a negative number (like -2) the argument of the function will have a "+" e.g. x+2.
These transformations are called horizontal shifts. In #6, the transformations were horizontal
shifts of right 2 and left 3. In #7, the transformations were vertical shifts of right 1 and left 4.
#9-10: Using the standard viewing window (ZOOM 6), graph the given parent function. Then
graph each of the given functions. (You should have only two graphs at a time.) Sketch the
graph and describe how the parent function could be moved to create the new function.
9. Parent function: f (x) = x 2
f (x) = (x − 4) 2
PreCalculus Standards 4 and 5
Functions and Graphs
f ( x) = ( x + 1) 2
Page 53 of 73
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10. Parent function: f (x) = x
Functions and Graphs –C
f (x) = x + 3
f (x) = x − 5
Consider the graph of f (x) = (x − 2)2 + 4 . The parent function is f (x) = x 2 and there are two
transformations, a horizontal shift of 2 to the right, and a vertical shift of up 4. The vertex of the
parabola is at (2, 4).
Identify the parent function and the transformations for the following functions. Sketch the
graph without using a calculator.
11. f (x) =
1
+1
x+2
12. f ( x) = x − 3 − 2
13. f (x) = x + 4 + 2
PreCalculus Standards 4 and 5
Functions and Graphs
Page 54 of 73
Columbus Public Schools 7/20/05
Transformations
Answer Key
Functions and Graphs –C
#1-2: Using the standard viewing window (ZOOM 6), graph the given parent function. Then
graph each of the given functions. (You should have only two graphs at a time.) Sketch the
graph and describe how the parent function could be moved to create the new function.
1. Parent function: f (x) = x 2
f (x) = x 2 + 2
f (x) = x 2 − 3
2. Parent function:
f (x) = x − 4
f (x) = x + 5
3. In general, if you have a parent function, y = f (x) , how will the graph of y = f (x) + c be
related to the parent function if c is positive? if c is negative?
The graph will move up c units.
The graph will move down c units.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 55 of 73
Columbus Public Schools 7/20/05
Functions and Graphs –C
When you start with a parent function and change it, it is called a transformation. These
transformations are called vertical shifts. In #1, the transformations were vertical shifts of up 2
and down 3. In #2, the transformations were vertical shifts of down 4 and up 5.
4. Without using a calculator, sketch the graph of f (x) = x 3 . Then sketch the graph of
f (x) = x 3 − 3 . Check your answer by graphing on your calculator. What kind of transformation
is this?
This is a vertical shift down
three.
5. Without using a calculator, sketch the graph of f (x) =
1
. Then sketch the graph of
x
1
+ 2 . Check your answer by graphing on your calculator. What kind of
x
transformation is this?
f (x) = f (x) =
This is a vertical shift up two.
#6-7: Using the standard viewing window (ZOOM 6), graph the given parent function. Then
graph each of the given functions. (You should have only two graphs at a time.) Sketch the
graph and describe how the parent function could be moved to create the new function.
6. Parent function: f (x) = x 3
f (x) = (x − 2)3
It moved 2 units right.
PreCalculus Standards 4 and 5
Functions and Graphs
f (x) = (x + 3)3
It moved 3 units left.
Page 56 of 73
Columbus Public Schools 7/20/05
7. Parent function: f (x) = x
f (x) = (x − 1)
It moved 1 unit right.
Functions and Graphs –C
f (x) = (x + 4)
It moved 4 units left.
8. In general, if you have a parent function, y = f (x) , how will the graph of y = f (x − b) be
related to the parent function if b is positive? The graph will move to the right.
In general, if you have a parent function, y = f (x) , how will the graph of y = f (x − b) be related
to the parent function if f b is negative? The graph will move to the left.
Notice that if b is a positive number (like 3) the argument of the function will have a "-" e.g. x-3.
If b is a negative number (like -2) the argument of the function will have a "+" e.g. x+2.
These transformations are called horizontal shifts. In #6, the transformations were horizontal
shifts of right 2 and left 3. In #7, the transformations were vertical shifts of right 1 and left 4.
#9-10: Using the standard viewing window (ZOOM 6), graph the given parent function. Then
graph each of the given functions. (You should have only two graphs at a time.) Sketch the
graph and describe how the parent function could be moved to create the new function.
9. Parent function: f (x) = x 2
f (x) = (x − 4) 2
It moved 4 units right.
PreCalculus Standards 4 and 5
Functions and Graphs
f ( x) = ( x + 1) 2
It moved 1 unit left.
Page 57 of 73
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10. . Parent function: f (x) = x
Functions and Graphs –C
f (x) = x + 3
f (x) = x − 5
It moved 3 units left.
It moved 5 units right.
Consider the graph of f (x) = (x − 2)2 + 4 . The parent function is f (x) = x 2 and there are two
transformations, a horizontal shift of 2 to the right and a vertical shift of up 4. The vertex of the
parabola is at (2,4).
Identify the parent function and the transformations for the following functions. Sketch the
graph without using a calculator.
1
+1
x+2
f ( x ) = 1 It moves 2 units left and
x
up 1 unit.
11. f (x) =
12. f ( x ) = x − 3 − 2
f ( x) = x
It moves 3 unit right and
down two units.
13. f (x) = x + 4 + 2
f ( x ) = x It moves 4 units
left and 2 units up.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 58 of 73
Columbus Public Schools 7/20/05
Introduction to Symmetry
Functions and Graphs –E
Name
Graphically, symmetry means that two objects are equally
displaced about a point or line. Two points are symmetric
about the y-axis if the y-axis is the perpendicular bisector of
the line connecting them. Put (2, 4) and (-2, 4) on the
coordinate plane at the left. Draw the line segment
connecting them. Repeat for (1, 1) and (1,-1). What would
be the other ordered pair needed to match up with (-3, 9)?
The graph of a function or relation is symmetric about the yaxis if every point on the graph has such a match. (The
origin is its own image.) Look at the graph of y = x2. Even
though it is not possible to test every point, it is obvious that
every point has an image across the y-axis. On the
coordinate plane above, sketch the graph of another function
that you believe to be symmetric about the y-axis.
One way to check to see if a graph is symmetric about the y-axis is to fold the graph about the yaxis. If all of the points from one side fall on the points from the other side, then the graph is
symmetric about the y-axis.
On your calculator, graph any equation of the form y = axn ,
where n is an even number, and sketch the graph on the
right. Compare your graph with other students. What did
you find?
This is the reason that functions whose graphs are
symmetric about the y-axis are called even functions. A
polynomial with all even exponents is symmetric about the
y-axis. There are other kinds of even functions as well, e.g.
f (x) = cos x .
It is possible to test whether a function is an even function without graphing by remembering that
opposite x-coordinates produce the same y-coordinate. In other words, f (x ) = f (-x ). To test a
function to see if it is even, replace x with –x and simplify. If the equation simplifies to the
original, it is symmetric about the y-axis and it is even. For example, with y = x2:
y = x2
Replace x with –x: y = (-x)2
Simplify
(-x)2= x2
And
y = x2
Try this with y = 3x4 + 2x2 -5 and then check graphically. This is an even function because all of
its exponents are even and any negatives will be raised to an even power. The last term, the 5, is
really 5x0 and so its exponent is also even.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 59 of 73
Columbus Public Schools 7/20/05
Look at a function whose exponents are all odd, like f (x) = x3.
Functions and Graphs –E
Each point has an image but it is a point that is
across the origin. In this case, the origin bisects
the line connecting the two images. This graph is
symmetric about the origin. Graph y = 4x3- 2x.
Notice that each point has an image across the
origin. It would be possible to verify the
symmetry with folding by folding along the x-axis
and then the y-axis. Find some other functions
which are symmetric about the origin.
Why are functions which are symmetric about the
origin called odd functions?
Remember that other kinds of functions may be odd as well. One example is y = cos x.
Notice that opposite x-coordinates produced opposite y-coordinates. In other words
f (x ) = - f (-x ). Use the method from the even functions to test your odd functions.
There is no name for relations which are symmetric about the x-axis. (They can’t be functions.
Why not?) One example would be x = y2. Sketch the graph and describe the folding which
would verify that the graph is symmetric to the x-axis. How could you verify this algebraically?
PreCalculus Standards 4 and 5
Functions and Graphs
Page 60 of 73
Columbus Public Schools 7/20/05
Introduction to Symmetry
Answer Key
Functions and Graphs –E
Graphically, symmetry means that two objects are equally
displaced about a point or line. Two points are symmetric
about the y-axis if the y-axis is the perpendicular bisector of
the line connecting them. Put (2, 4) and (-2, 4) on the
coordinate plane at the left. Draw the line segment
connecting them. Repeat for (1, 1) and (1,-1). What would
be the other ordered pair needed to match up with (-3, 9)?
(3, 9)
The graph of a function or relation is symmetric about the yaxis if every point on the graph has such a match. (The
origin is its own image.) Look at the graph of y = x2. Even
though it is not possible to test every point, it is obvious that
every point has an image across the y-axis. On the
coordinate plane above, sketch the graph of another function
that you believe to be symmetric about the y-axis.
Answers will vary.
One way to check to see if a graph is symmetric about the yaxis is to fold the graph about the y-axis. If all of the points
from one side fall on the points from the other side, then the
graph is symmetric about the y-axis.
On your calculator, graph any equation of the form y = axn ,
where n is an even number, and sketch the graph on the
right. Compare your graph with other students. What did
you find?
This is the reason that functions whose graphs are
symmetric about the y-axis are called even functions. A
polynomial with all even exponents is symmetric about the
y-axis. There are other kinds of even functions as well, e.g.
f (x ) = cos x .
It is possible to test whether a function is an even function without graphing by remembering that
opposite x-coordinates produce the same y-coordinate. In other words, f (x) = f (-x ). To test a
function to see if it is even, replace x with –x and simplify. If the equation simplifies to the
original, it is symmetric about the y-axis and it is even. For example, with y = x2:
y = x2
Replace x with –x: y = (-x)2
Simplify
(-x)2= x2
And
y = x2
Try this with y = 3x4 + 2x2 -5 and then check graphically. This is an even function because all of
its exponents are even and any negatives will be raised to an even power. The last term, the 5, is
really 5x0 and so its exponent is also even.
3x4 + 2x2 -5 = 3(-x4 )+ 2(-x)2 -5
PreCalculus Standards 4 and 5
Functions and Graphs
Page 61 of 73
Columbus Public Schools 7/20/05
Look at a function whose exponents are all odd, like f (x ) =x3.
Functions and Graphs –E
Each point has an image but it is a point that is
across the origin. In this case, the origin bisects
the line connecting the two images. This graph is
symmetric about the origin. Graph y = 4x3- 2x.
Notice that each point has an image across the
origin. It would be possible to verify the
symmetry with folding by folding along the x-axis
and then the y-axis. Find some other functions
which are symmetric about the origin.
Answers will vary.
Why are functions which are symmetric about the
origin called odd functions?
In a polynomial, all of the exponents are odd.
Remember that other kinds of functions may be
odd as well. One example is y = cos x.
Notice that opposite x-coordinates produced opposite y-coordinates. In other words
f (x ) = - f (-x ). Use the method from the even functions to test your odd functions.
Answers will vary.
There is no name for relations which are symmetric about the x-axis. (They can’t be functions.
Why not?) One example would be x = y2. Sketch the graph and describe the folding which
would verify that the graph is symmetric to the x-axis. How could you verify this algebraically?
Answers will vary.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 62 of 73
Columbus Public Schools 7/20/05
The Mile Run
Functions and Graphs –F
Name
The Mile Run chart (at the end of this activity) gives a list of records for the Mile Run from 1911
to 1999.
1. Use the records from 1911 to 1945.
a. Using 1900 for t = 0, enter the year into a L1 and the time in seconds in L2. Make a
scatterplot with the year on the x-axis and the time in seconds on the y-axis. Lay a piece of
spaghetti over the calculator screen to find a line where the data points are evenly distributed
on either side of the spaghetti. Choose two data points which are close to this line and use
them to estimate the equation of the line.
Sketch your scatterplot below.
and the equation of the spaghetti line.
b. Give your two points
c. Use your calculator to graph the spaghetti line with the statplot. On the sketch above,
circle your two points and sketch the line.
d. Using the equation of your line, estimate the mile run record for 1962
and 1985
.
1980
e. Use your calculator to find the equation of the line of best fit by using the linear regression
function.
f. Graph it with the data and the spaghetti line.
g. To compare the predictions of the two lines with the actual times, complete the chart.
Date
Prediction from
Spaghetti Line
Prediction from
Line of Best Fit
Actual Record
Prediction from Line of
Best Fit-Actual Record
1915
1933
1937
1944
PreCalculus Standards 4 and 5
Functions and Graphs
Page 63 of 73
Columbus Public Schools 7/20/05
Functions and Graphs –F
h. The predictions in the chart above are examples of interpolation, because the years were
within the period of time when the data were collected.
What is the slope of the line of best fit?
i. What does the slope tell you about how the record for the mile run was changing during
this period?
j. What is the y-intercept of the line of best fit?
in terms of the data?
What is its meaning
k. Use the equation of the line of best fit to complete the chart below. Use the data in from
the Mile Run chart to complete the actual time.
Date
Prediction from Line of Best
Fit
Actual Time
Prediction minus
Actual time
1962
1980
1985
The predictions on this chart are examples of extrapolation, because the years were outside
the period of time when the data was collected.
l. Use the line to predict the year predicted by the line of best fit when the four-minute mile
would have been run.
2. Use the records from 1954-1999.
a. In L3, enter the years from 1954-1999, again using 1900 as t = 0 and make a scatterplot.
Enter the time in seconds into L4. Use the linear regression function on your calculator
to find the line of best fit for this data and graph it along with your data. Give the
equation here. Round the numbers to three decimal places for your answer, but leave the
original numbers in the Y= menu on the calculator.
When was the record
b. What is the slope of the 1954-1999 line of best fit?
changing faster, during 1911-1945 or 1954-1999? How did you know?
PreCalculus Standards 4 and 5
Functions and Graphs
Page 64 of 73
Columbus Public Schools 7/20/05
c. What is the y-intercept and what does it mean in terms of the data?
Functions and Graphs –F
d. Use the 1954-1999 equation to predict when a 3:30-mile will be run.
3. Use the records from 1911-1999.
a. Use augment (L1, L3) STO L5 and augment (L2, L4) STO L6 to put all of the years
together in L5 and all of times in L6. Make a scatterplot, and find the equation of the line
of best fit. Give the equation here.
b. What are the slope and y-intercept? What do they mean in terms of the data?
c. Which of the three lines of best fit show the record decreasing the fastest?
d. Which of the three lines of best fit show the record decreasing the slowest?
e. Use the TABLE and ASK feature on your calculator to complete the chart below.
Date
1911-1945 prediction
1954-1999 prediction
1911-1999 prediction
Actual
Record
1913
1933
1945
1957
1980
f. Which interval (1911-1945, 1954, or 1911-1999) gave the equation that made the best
prediction for each of the years above? Which interval made the worst prediction?
I = interpolation and E = extrapolation
Date
1913
1933
1945
1957
1980
Best (I or E)
Worst (I or E)
g. Use the 1911-1999 line to estimate the time for the mile run when Julius Caesar was
assassinated. (LOOK IT UP!!)
h. Use the 1911-1999 line to estimate the time for the mile run in the year 4000 AD.
i. What do you think about the reliability of predictions made using interpolation and
extrapolation? Why is the use of extrapolation sometimes misleading? Support your
views with examples from this exercise. Also, you might want to think about extreme
situations.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 65 of 73
Columbus Public Schools 7/20/05
Mile Run Chart
Functions and Graphs –F
Time
Athlete
Country
Year
Location
4:15.4
John Paul Jones
United States
1911
Cambridge, Mass.
4:14.4
John Paul Jones
United States
1913
Cambridge, Mass
4:12.6
Norman Taber
United States
1915
Cambridge, Mass
4:10.4
Paavo Nurmi
Finland
1923
Stockholm
4:09.2
Jules Ladoumegue
France
1931
Paris
4:07.6
Jack Lovelock
New Zealand
1933
Princeton, N.J.
4:06.8
Glenn Cunningham
United States
1934
Princeton, N.J.
4:06.4
Sydney Wooderson
England
1937
London
4:06.2
Gundar Hägg
Sweden
1942
Goteborg, Sweden
4:06.2
Arne Andersson
Sweden
1942
Stockholm
4:04.6
4:02.6
Gundar Hägg
Arne Andersson
Sweden
Sweden
1942
1943
Stockhom
Goteborg, Stockholm
4:01.6
Arne Andersson
Sweden
1944
Malmo, Sweden
4:01.4
Gundar Hägg
Sweden
1945
Malmo, Sweden
3:59.4
Roger Bannister
England
1954
Oxford, England
3:58.0
John Landy
Australia
1954
Turku, Findland
3:57.2
Derek Ibbotson
England
1957
London
3:54.5
Herb Elliott
Australia
1958
Dublin
3:54.4
Peter Snell
New Zealand
1962
Wanganui, N.Z.
3:54.1
Peter Snell
New Zealand
1964
Auckland, N.Z.
3:53.6
Michel Jazy
France
1965
Rennes, France
3:51.3
Jim Ryun
United States
1966
Berkeley, Calif.
3:51.1
Jim Ryun
United States
1967
Bakersfield, Calif.
3:51.0
Filbert Bayi
Tanzania
1975
Kingston, Jamaica
3:49.4
John Walker
New Zealand
1975
Goteborg, Sweden
3:49.0
Sebastian Coe
England
1979
Oslo
3:48.8
Steve Ovett
England
1980
Oslo
3:48.53
Sebastian Coe
England
1981
Zurich, Switzerland
3:48.40
Steve Ovett
England
1981
Koblenz, W. Ger
3:47.33
Sebastian Coe
England
1981
Brussels
3:46.31
Steve Cram
England
1985
Oslo
3:44.39
Noureddine Morceli
Algeria
1993
Rieti, Italy
3:43.13
Hicham El Guerrouj
Morocco
1999
Rome, Italy
PreCalculus Standards 4 and 5
Functions and Graphs
Page 66 of 73
Columbus Public Schools 7/20/05
The Mile Run
Answer Key
Functions and Graphs –F
The Mile Run Chart (at the end of this activity) gives a list of records for the Mile Run from
1911 to 1999.
1. Use the records from 1911 to 1945.
a. Using 1900 for t=0, enter the year into a L1 and the time in seconds in L2. Make a
scatterplot with the year on the x-axis and the time in seconds on the y-axis. Lay a piece
of spaghetti over the calculator screen to find a line where the data points are evenly
distributed on either side of the spaghetti. Choose two data points which are close to this
line and use them to estimate the equation of the line.
Sketch your scatterplot below.
and the equation of the
(42, 244.6)
b. Give your two points. (11, 255.4)
spaghetti line. y= -.348x+259.232 (Answers may vary.)
c. Use your calculator to graph the spaghetti line with the statplot. On the sketch above,
circle your two points and sketch the line. (Answers may vary)
See above.
d. Using the equation of your line, estimate the mile run record for 1962 237.66 1980
231.39 and 1985
229.65 .(Answers may vary)
e. Use your calculator to find the equation of the line of best fit by using the linear
regression function.
y(x)=-.431859x + 262.207
f. Graph it with the data and the spaghetti line. (See above)
g. To compare the predictions of the two lines with the actual times, complete the chart.
Date
1915
Prediction from
Spaghetti Line
254.012
Prediction from
Line of Best Fit
255.729
Actual
Record
252.6
Prediction from Line of
Best Fit-Actual Record
3.129
1933
247.748
247.955
247.
.955
1937
246.356
246.228
246.4
-.172
1944
243.92
243.205
241.6
1.605
PreCalculus Standards 4 and 5
Functions and Graphs
Page 67 of 73
Columbus Public Schools 7/20/05
Functions and Graphs –F
h. The predictions in the chart above are examples of interpolation, because the years were
within the period of time when the data was collected.
What is the slope of the line of best fit?
-.431859
i. What does the slope tell you about how the record for the mile run was changing during
this period? It was decreasing at a rate of -.431859 seconds per year.
262.207
What is its
j. What is the y-intercept of the line of best fit?
meaning in terms of the data? It is the predicted number of seconds for the record in
1900.
k. Use the equation of the line of best fit to complete the chart below. Use the data from the
Mile Run Chart to complete the actual time.
Date
1962
Prediction from Line of Best
Fit
235.432
Actual Time
234.4
Prediction minus
Actual time
1.032
1980
227.658
228.8
-1.142
1985
225.499
226.31
-1.011
The predictions on this chart are examples of extrapolation, because the years were outside
the period of time when the data was collected.
l. Use the line to predict the year predicted by the line of best fit when the four-minute mile
would have been run.
1951
2. Use the record from 1954-1999.
a. In L3, enter the years from Chart II, again using 1900 as t=0 and make a scatterplot.
Enter the time in seconds into L4. Use the linear regression function on your calculator
to find the line of best fit for this data and graph it along with your data. Give the
equation here. Round the numbers to three decimal places for your answer, but leave the
original numbers in the Y= menu on the calculator. y=336x + 255.393
b. What is the slope of the 1954-1999 line of best fit? -.336 When was the record
changing faster, during 1911-1945 or 1954-1999? How did you know? 1911-1945. In
1911-1945, it was changing at .432 per year and in 1954-1999, it was changing at
.336 seconds per year.
c. What is the y-intercept and what does it mean in terms of the data?
255.393. It is the time predicted for 1900 by this equation.
PreCalculus Standards 4 and 5
Functions and Graphs
Page 68 of 73
Columbus Public Schools 7/20/05
d. Use the 1954-1999 equation to predict when a 3:30 mile will be run. 2035 Functions and Graphs –F
3. Use the records from 1911-1999.
a. Use augment (L1,L3) STO L5 and augment (L2,L4) STO L6 to put all of the years
together in L5 and all of times in L6. Make a scatterplot, and find the equation of the line
of best fit. Give the equation here.
y = -413x + 261.249
b. What are the slope and y-intercept? What do they mean in terms of the data?
Slope is -.413, the number of seconds per year that the time decreased.
y-intercept is 261.249, the predicted time for the mile run in 1900.
c. Which of the three lines of best fit show the record decreasing the fastest?
Fastest 1911-1945
d. Which of the three lines of best fit show the record decreasing the slowest?
Slowest 1945-1999
e. Use the TABLE and ASK feature on your calculator to complete the chart below.
Date
1911-1945 prediction 1954-1999 prediction 1911-1999 prediction Actual
Record
1913
256.593
251.022
255.875
254.4
1933
247.955
244.299
247.607
247.6
1945
242.773
240.265
242.646
241.4
1957
237.591
236.231
237.685
237.2
1980
227.658
228.499
228.177
228.8
f. Which interval of years (1911-1945, 1954-1999, or 1911-1999) gave the equation that
made the best prediction for each of the years above? Which interval made the worst
prediction?
I = interpolation and E = extrapolation
Date
1913
1933
1945
1957
1980
Best (I or E)
1911-1945 I
1911-1933 I
1954-1999 E
1911-1933 I
1954-1999 I
Worst (I or E)
1954-1999 E
1954-1999 E
1911-1945 I
1954-1999 I
1911-1945 E
g. Use the 1911-1999 line to estimate the time for the mile run when Julius Caesar was
assassinated. (LOOK IT UP!!)
He was assassinated in 44BC. The predicted time is 1064.88 sec. (17 hours and 45
min.)
h. Use the 1911-1999 line to estimate the time for the mile run in the year 4000 AD.
-606.87
i. What do you think about the reliability of predictions made using interpolation and you
extrapolation? Why is the use of extrapolation sometimes misleading? Support your
views with examples from this exercise. Also, you might want to think about extreme
situations. In general, interpolation is more reliable than extrapolation.
Interpolation is most accurate towards the middle of the data set and may be not as
accurate at the edges of the data set (see 1945) Also, interpolation is only reliable
when the data is fairly close to the data set. It could not have taken over 17 hours to
run a mile in 44BC, nor do we expect time travel in the year 4000AD.
PreCalculus Standards 4 and 5
Columbus Public Schools 7/20/05
Page 69 of 73
Functions and Graphs
Concepts in Graphical Analysis Functions and Graphs –Reteach
Name
1. Given the graph below:
(-1,11)
(-3,6)
(0, 10)
(1.8, 0)
(4.7, 0)
(3.5, -10)
Find:
a. x-intercepts
b. y-intercepts
c. local maxima
d. local minima
e. Domain
f. Range
g. interval where increasing
h. interval where decreasing
PreCalculus Standards 4 and 5
Functions and Graphs
Page 70 of 73
Columbus Public Schools 7/20/05
2. For this graph, YOU supply the points…
Functions and Graphs –Reteach
Find:
a. x-intercepts
b. y-intercepts
c. local maxima
d. local minima
e. Domain
f. Range
g. interval where increasing
h. interval where decreasing
PreCalculus Standards 4 and 5
Functions and Graphs
Page 71 of 73
Columbus Public Schools 7/20/05
Concepts in Graphical Analysis Functions and Graphs –Reteach
Answer Key
1. Given the graph below:
(-1,11)
(-3,6)
(0, 10)
(1.8, 0)
(4.7, 0)
(3.5, -10)
Answers may vary slightly
Find:
a. x-intercepts
(1.8, 0) and (4.7,0)
b. y-intercepts
(0,10)
c. local maxima
(-1, 11)
d. local minima
(-3, 6) and (3.5,-10)
e. Domain
(- , )
f. Range
[-10, )
g. interval where increasing
(-3,-1) (3.5, )
h. interval where decreasing
(- ,-3) (-1, 3.5)
PreCalculus Standards 4 and 5
Functions and Graphs
Page 72 of 73
Columbus Public Schools 7/20/05
2. For this graph, YOU supply the points…
Functions and Graphs –Reteach
Find:
a. x-intercepts
(-3, 0), (1, 0), (3, 0) and (5, 0)
b. y-intercepts
(0, 4)
c. local maxima
(-1.25, 8), (3, 0)
d. local minima
(-3, 0), (1.5,-1), (4.5,-2.5)
e. Domain
(-∞,∞)
f. Range
[-2.5, ∞)
g. interval where increasing
(-3,-1.25) ∪ (1.5, 3) ∪ (4.5,∞)
h. interval where decreasing
(-∞,-3) ∪ (-1.25, 1.5) ∪ (3, 4.5)
PreCalculus Standards 4 and 5
Functions and Graphs
Page 73 of 73
Columbus Public Schools 7/20/05
COLUMBUS PUBLIC SCHOOLS
MATHEMATICS CURRICULUM GUIDE
SUBJECT
PreCalculus
STATE STANDARDS 4 and 5
Patterns, Functions, and Algebra;
Data Analysis and Probability
TIME RANGE
40-45 days
GRADING
PERIOD
1-2
MATHEMATICAL TOPIC 2
Polynomial, Power, and Rational Functions
A)
B)
C)
D)
CPS LEARNING GOALS
Determines the characteristics of the polynomial functions of any degree, general shape,
number of real and nonreal (real and nonreal), domain and range, and end behavior, and finds
real and nonreal zeros.
Identifies power functions and direct and inverse variation.
Describes and compares the characteristics of rational functions; e.g., general shape, number
of zeros (real and nonreal), domain and range, asymptotic behavior, and end behavior.
Analyzes and interprets bivariate data to identify patterns, note trends, draw conclusions, and
make predictions.
COURSE LEVEL INDICATORS
Course Level Indicators (i.e., How does a student demonstrate mastery?):
9 Identifies the intervals on which a polynomial is increasing or decreasing and determines
maxima and minima graphically (using technology). Math A:11-A:04
9 Uses limit notation to describe asymptotic and end behaviors. Math MP:11/12-H and
Math A:11-A:03
9 Models real world data with polynomial, power, and rational functions. Math A:11-A:03
and Math D:11-A:04
9 Explains the relationship between the Fundamental Theorem of Algebra, the graph of a
polynomial function, and the factors of the polynomial over the set of complex numbers.
Math A:11-A:03
9 Identifies power functions and their graphs. Math A:11-A:03
9 Identifies the maximum and minimum points of polynomial and rational functions
graphically and with technology. Math A:11-A:04
9 Analyzes end behavior of a function using limits. Math A:11-A:03
9 Explains the relationship between zeros, factors, and x-intercepts of the graph of a
polynomial function, including the use of the Remainder, Factor, and Rational Zeros
Theorems. Math A:11-A:03
9 Uses synthetic division to evaluate polynomials and find factors of polynomials.
Math A:11-A:03
9 Determines horizontal, vertical, and slant asymptotes of rational functions. Math A:11-A:03
9 Determines points of discontinuity and intervals on which a function is continuous.
Math A:11-A:03
9 Connects geometric transformations on the graph of a polynomial or rational function to
changes of parameters in an equation. Math A:11-A:03
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 1 of 71
Columbus Public Schools 7/20/05
Previous Level:
9 Identifies polynomial functions. Math A:11-A:03
9 Identifies rational functions. Math A:11-A:03
9 Determines the domain and range of a rational function. Math A:11-A:03
9 Solves equations involving radical expressions and complex roots. Math A:11-A:08
9 Understands the geometric representation of complex numbers and the absolute value of
complex numbers. Math A:11-A:08
Next Level:
9 Analyzes polynomials and rational functions by investigating rates of change. Math A:12A:10
9 Finds area and volume of regions and solids defined by polynomials and rational functions.
Math A:12-A:08
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 2 of 71
Columbus Public Schools 7/20/05
The description from the state, for the Patterns, Functions, and Algebra Standard says:
Students use patterns, relations, and functions to model, represent and analyze problem situations
that involve variable quantities. Students analyze, model and solve problems using various
representations such as tables, graphs, and equations
The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is:
A. Analyze functions by investigating rates of change, intercepts, zeros, asymptotes and local
and global behavior.
The description from the state, for the Data Analysis and Probability Standard says:
Students pose questions and collect, organize represent, interpret and analyze data to answer
those questions. Students develop and evaluate inferences, predictions and arguments that are
based on data.
The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is:
A. Create and analyze tabular and graphical displays of data using appropriate tools, including
spreadsheets and graphing calculators.
The description from the state, for the Mathematical Processes Standard says:
Students use mathematical processes and knowledge to solve problems. Students apply
problem-solving and decision-making techniques, and communicate mathematical ideas.
The grade-band benchmark from the state, for this topic in the grade band 11 – 12 is:
H. Use formal mathematical language and notation to represent ideas, to demonstrate
relationships within and among representation systems, and to formulate generalizations.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 3 of 71
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Polynomial, Power, Rational- A
Given the function f(x) = ax4 + bx3 + cx2 + dx + e, where a, b, c, d, and e are real numbers, which
could be a correct statement about the zeros of the polynomial?
A. There are one real and three nonreal zeros.
B. There are three real and one nonreal zeros
C. There are one real and two nonreal zeros.
D. There are two real and two nonreal zeros.
Suppose that a polynomial function f is defined in such a way that f (-2.8) = -8 and f (4) = 9.
What conclusion does the Intermediate Value Theorem allow you to draw?
A. There must be two zeros between 2.8 and -4.
B. There must be a zero between -8 and 9.
C. There must be a zero between 8 and -9.
D. There must be a zero between -2.8 and 4.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 4 of 71
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Polynomial, Power, Rational- A
Answers/Rubrics
Low Complexity
Given the function f(x) = ax4 + bx3 + cx2 + dx + e, where a, b, c, d, and e are real numbers,
which could be a correct statement about the zeros of the polynomial?
A. There are one real and three nonreal zeros.
B. There are three real and one nonreal zeros
C. There are one real and two nonreal zeros.
D. There are two real and two nonreal zeros.
Answer: D
Moderate Complexity
Suppose that a polynomial function f is defined in such a way that f (-2.8) = -8 and f (4) = 9.
What conclusion does the Intermediate Value Theorem allow you to draw?
A. There must be two zeros between 2.8 and -4.
B. There must be a zero between -8 and 9.
C. There must be a zero between 8 and -9.
D. There must be a zero between -2.8 and 4.
Answer: D
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 5 of 71
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Polynomial, Power, Rational- A
i is a zero of f(x) = x4 – 6x3 + 7x2 – 6x + 6. What are the other zeros?
A. - i, 3 + 2 3 , 3 − 2 3
B. –i, 3 + 3 , 3 − 3
C. –i, 1+ 3 , 1− 3
D. –i, −3 + 3 , −3 − 3
Given the graph of the sixth degree polynomial below:
100
80
60
Describe the zeros of the polynomial including their multiplicities.
40
20
-5 -4 -3 -2 -1
1
2
3
4
5
-20
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 6 of 71
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Polynomial, Power, Rational- A
Answers/Rubrics
High Complexity
i is a zero of f(x) = x4 – 6x3 + 7x2 – 6x + 6. What are the other zeros?
A. -i, 3 + 2 3 , 3 − 2 3
B. –i, 3 + 3 , 3 − 3
C. –i, 1+ 3 , 1− 3
D. –i, −3 + 3 , −3 − 3
Answer: B
Short Answer/Extended Response
Given the graph of the sixth degree polynomial below:
100
80
60
Describe the zeros of the polynomial including their multiplicities.
40
20
-5 -4 -3 -2 -1
1
2
3
4
5
-20
Answer: -3 is a zero of multiplicity one,
-1 is a zero of multiplicity two, and 2 is a zero of multiplicity 3.
A 2-point response correctly identifies all zeros and their multiplicities.
A 1-point response correctly identifies the zeros with errors in multiplicity.
A 0-point response shows no mathematical understanding.
(Teacher Note: Although there are other combinations of zeros, students should
understand that the changes in concavity suggest this configuration.)
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 7 of 71
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Polynomial, Power, Rational - B
Which of the following is a power function?
A. f(x) = 2x3
B. f(x) = 2x3 + 5
C. f ( x) = 3x
D. f ( x) = 3x + 4
Which statement expresses the statement "The surface area of a sphere S varies directly as the
square of its radius r," as a power function?
k
A) S = 2
r
B) S = k 2r
C) S = kr 2
D) S =
r2
k
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 8 of 71
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Polynomial, Power, Rational - B
Answers/Rubrics
Low Complexity
Which of the following is a power function?
A. f(x) = 2x3
B. f(x) = 2x3 + 5
C. f ( x) = 3x
D. f ( x) = 3x + 4
Answer: A
Moderate Complexity
Which statement expresses the statement "The surface area of a sphere, S, varies directly as the
square of its radius, r," as a power function?
k
A) S = 2
r
B) S = k 2r
C) S = kr 2
D) S =
r2
k
Answer: C
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 9 of 71
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Polynomial, Power, Rational - B
The table below shows the average distances and orbit periods for the six innermost planets:
Planet
Average Distance from Sun
Period of orbit (in days)
(in millions of km)
Mercury
57.9
88
Venus
108.2
224
Earth
149.6
365.2
Mars
227.9
687
Jupiter
778.3
4332
Saturn
1427
10,760
Find a power function to model orbital period as a function of average distance from the Sun.
Neptune is 4497 million kilometers from the Sun on average. Which of the following is the
orbital period for Neptune predicted by the model?
A. 20,534 days
B. 60,313 days
C. 80,676 days
D. 120,495 days
A power function is of the form f ( x) = kx a . Look at the graph and describe what you know
about the values for k and a and support your answers.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 10 of 71
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Polynomial, Power, Rational - B
Answers/Rubrics
High Complexity
The table below shows the average distances and orbit periods for the six innermost planets:
Planet
Average Distance from Sun
Period of orbit (in days)
(in millions of km)
Mercury
57.9
88
Venus
108.2
224
Earth
149.6
365.2
Mars
227.9
687
Jupiter
778.3
4332
Saturn
1427
10,760
Find a power function to model orbital period as a function of average distance from the Sun.
Neptune is 4497 million kilometers from the Sun on average. Which of the following is the
orbital period for Neptune predicted by the model?
A. 20,534 days
B. 60,313 days
C. 80,676 days
D. 120,495 days
Answer: B
Short Answer/Extended Response
A power function is of the form f ( x) = kx a . Look at the graph and describe what you know
about the values for k and a and support your answers.
Answer: k is negative because the graph is reflected
about the x-axis. a is negative because the y-axis is a
vertical asymptote and a can be written as a fraction
with denominator a power of two because there is only
one branch of the curve.
A 4-point response indicates that k and a are negative,
and that a can be written as a fraction where the denominator is a power of two and
supports these answers.
A 3-point response indicates that k and a are negative and supports these answers.
A 2-point response indicates that k and a are negative with no support.
A 1-point response indicates that either k or a is negative with no support.
A 0-point response shows no mathematical understanding.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 11 of 71
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
3x 5 + 4 x 4 + 2 x 2 − 1
, for x ≠ -2?
x+2
Which function is equivalent to f ( x ) =
A. g(x) = 3x 4 + 2x 3 + 4x 2 + 8x −
15
x+2
B. g(x) = 3x 4 − 2x 3 + 4x 2 − 6x + 12 −
C. g(x) = 3x 4 − 2x 3 + 4x 2 + 6 −
25
x+2
13
x+2
D. g(x) = 3x 4 − 2x 3 + 6x 2 − 12 +
Given the function f ( x ) =
Polynomial, Power, Rational - C
23
x+2
3x 2 − 9 x − 5
, which statement is true?
4 x2 − 7x + 7
A. There are no asymptotes.
B.
y = 0 is a horizontal asymptote.
C.
y = 3/4 is a horizontal asymptote.
D.
y = 3x + 4 is a slant asymptote.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 12 of 71
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Polynomial, Power, Rational - C
Answers/Rubrics
Low Complexity
3x 5 + 4x 4 + 2x 2 − 1
, for x ≠ -2?
x+2
Which function is equivalent to f (x) =
A. g(x) = 3x 4 + 2x 3 + 4x 2 + 8x −
15
x+2
B. g(x) = 3x 4 − 2x 3 + 4x 2 − 6x + 12 −
C. g(x) = 3x 4 − 2x 3 + 4x 2 + 6 −
25
x+2
13
x+2
D. g(x) = 3x 4 − 2x 3 + 6x 2 − 12 +
23
x+2
Answer: B
Moderate Complexity
Given the function f (x) =
3x 2 − 9x − 5
, which statement is true?
4x 2 − 7x + 7
A.
There are no asymptotes.
B.
y = 0 is a horizontal asymptote.
C.
y = 3/4 is a horizontal asymptote.
D.
y = 3x + 4 is a slant asymptote.
Answer: C
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 13 of 71
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Polynomial, Power, Rational - C
Which equation correctly identifies the end behavior asymptote(s) of f (x) =
x 3 − 3x 2 + 3x + 1
?
x −1
A. x = 1
B. y = x – 1
C. y = x2 – 2x + 1
D. y = -3x3 + 3x + 1
2x 2 − 2
using a graphing calculator. Verify the x-intercepts,
x2 − 4
asymptotes, domain, and range algebraically.
Graph the function f (x) =
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 14 of 71
Columbus Public Schools 7/20/05
PRACTICE ASSESSMENT ITEMS
Polynomial, Power, Rational - C
Answers/Rubrics
High Complexity
Which equation correctly identifies the end behavior asymptote(s) of f (x) =
x 3 − 3x 2 + 3x + 1
?
x −1
A. x=1
B. y=x–1
C. y=x2–2x+1
D. y=-3x3+3x+1
Answer: C
Short Answer/Extended Response
2x 2 − 2
using a graphing calculator. Verify the x-intercepts,
x2 − 4
10
asymptotes, domain, and range algebraically.
8
Graph the function f (x) =
6
Answer:
4
2
-8 -7 -6 -5 -4 -3 -2 -1
-2
1 2 3 4 5 6 7 8
-4
-6
-8
-10
Setting the factors of the numerator equal to zero: 2(x+1)(x-1) = 0, the zeros are 1 and -1.
Setting the factors of the denominator equal to zero: (x+2)(x-2) = 0, the vertical asymptotes
are x = 2 and x = -2 and the domain is (−∞, −2 ) ∪ (−2, 2 ) ∪ (2, ∞) . Because the degree of the
numerator is equal to the degree of the denominator, the leading coefficients of the
numerator and denominator yield y = 2 as a horizontal asymptote and the range
is (−∞, 2 ) ∪ (2, ∞) .
A 2-point response correctly identifies the x-intercepts, the asymptotes, the domain, and the
range and supports them algebraically.
A 1-point response correctly identifies the x-intercepts, the asymptotes, the domain, and the
range but does not correctly support them algebraically.
A 0-point response shows no mathematical understanding.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 15 of 71
Columbus Public Schools 7/20/05
Teacher Introduction
Polynomial, Power, and Rational Functions
In topic one, students studied properties of functions in general and looked at the basic properties
of twelve functions. In this topic, students concentrate on three classes of functions--polynomial,
power, and rational.
The students will probably have had studied quadratic and cubic functions in the past. In this
topic, they will look at the properties common to all polynomial functions with particular
attention to zeros and factors. The applications of power functions are an important part of this
topic. Students have had very little exposure to the properties of rational functions, and the study
of their asymptotic and end behavior is especially important.
The strategies and activities section of learning goal A refer to teacher notes (included in this
Curriculum Guide) that provide you, the teacher, with a method of introducing these three
functions.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 16 of 71
Columbus Public Schools 7/20/05
TEACHING STRATEGIES/ACTIVITIES
Vocabulary: polynomial function, leading coefficient, zero function, constant function,
linear function, quadratic function, slant line, average rate of change, depreciation,
constant term, linear correlation, linear correlation coefficient, axis of symmetry, vertex,
standard quadratic form, vertex form, standard polynomial form, maximum, minimum,
extremum, vertical velocity, identity function, squaring function, least-squares line,
power function, constant of variation, proportional, direct variation, inverse variation,
monomial function, cubing function, square root function, term, quartic function,
multiplicity, repeated zero, Intermediate Value Theorem, polynomial interpolation,
quotient, remainder, dividend, divisor, Remainder Theorem, Factor Theorem, synthetic
division, Rational Zeros, Theorem, upper bound, lower bound, complex number,
imaginary part, real part, additive identity, additive inverse, complex conjugate,
multiplicative identity, multiplicative inverse, reciprocal, discriminant, complex plane,
real axis, imaginary axis, absolute value, modulus, distance, midpoint, Fundamental
Theorem of Algebra, Linear Factorization Theorem, rational function, reciprocal
function, limit, horizontal asymptote, end behavior, vertical asymptote, slant asymptote,
intercept, rational equation, extraneous solution, sign chart, inequality, oblique.
Core:
Learning Goal A: Determines the characteristics of the polynomial functions of any degree,
general shape, number of real and nonreal (real and nonreal), domain and range, and end
behavior, and finds real and nonreal zeros.
1. Emphasize that the correct use of vocabulary is essential. Students must understand how
zeros, x-intercepts, and factors are related and use the correct word to describe each. With
rational functions, look at specific examples and then generalize.
2. Do the activity "Patterns in Polynomial End-Behavior" (included in this Curriculum Guide).
Here students will discover the forms and shapes of the four basic types of polynomial
functions.
3. Students use algebraic and graphical methods to analyze a parabolic trajectory by completing
the activity "Home Run" (included in this Curriculum Guide).
4. The relationship between linear factors of polynomials and x-intercepts is explored in the
activity, "Graphs, Factors, Zeros" (included in this Curriculum Guide).
5. In the activity "Connections With Multiplicities" (included in this Curriculum Guide),
students explore how graphs behave at their x-intercepts depending on their multiplicity.
6. Students practice using different methods of factoring polynomials utilizing the Rational
Zero Theorem and the Factor Theorem in the activity "Factors, Remainders, Zeros" (included
in this Curriculum Guide).
Learning Goal B: Identifies power functions and direct and inverse variation.
1. Students compare various properties of the graphs of power functions and exponential
functions by completing the activity "Exponential vs. Power" (included in this Curriculum
Guide).
2. In the activity "Investigating Pendulum Length" (included in this Curriculum Guide) students
gather pendulum motion data and then use the Power Regression feature of the TI-83 to
model this data.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 17 of 71
Columbus Public Schools 7/20/05
Learning Goal C: Describes and compares the characteristics of rational functions; e.g.,
general shape, number of zeros (real and nonreal), domain and range, asymptotic behavior, and
end behavior.
1. Students use the formal definition of rational function to identify which functions are rational
in the activity "Rational Functions" (included in this Curriculum Guide).
2. Students discover the relationship between certain algebraic features of rational functions and
their graphical consequences by completing the activity "Graphing Rational Functions"
(included in this Curriculum Guide).
3. In the activity "End-Behavior of Rational Functions" (included in this Curriculum Guide),
students are asked to relate end behavior to the degree of the numerator and denominator.
4. Students practice determining where a function is positive and where it is negative in order to
assist with graphing by completing the activity "Sign Chart Analysis" (included in this
Curriculum Guide).
Learning Goal D: Analyzes and interprets bivariate data to identify patterns, note trends, draw
conclusions, and make predictions.
1. In the activity "Keep it Bouncing" (included in this Curriculum Guide), students use a CBR
and graphing calculator to examine the parabolic motion which models a bouncing ball.
Reteach:
1. Review linear equations, x-intercepts, y-intercepts.
2. Review quadratic equations, factoring, use of the quadratic formula.
3. In the activity "Adventures in Multiplicity" (included in this Curriculum Guide), students
review the relation between the multiplicity of a polynomial factor and the graph's behavior
at that zero.
Extension:
1. This is an extension to the pendulum activity. Design and conduct an experiment to find out
how the weight of a pendulum affects the period. You can do this experiment alone or with
your group. Type a two-page report (double space). Be sure to include a description on the
materials used, the collected data, how you conducted the experiment, graphical and/or
algebraic support on your conclusion regarding how the weight of a pendulum affects the
period.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 18 of 71
Columbus Public Schools 7/20/05
RESOURCES
Learning Goal A:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 162-180; 193236
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 31-32; 35-42
Learning Goal B:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 181-192.
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 33-34
Learning Goal C:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 237-273.
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 43-48
Learning Goal D:
Textbook: PreCalculus: graphical, numerical, algebraic, Pearson (2004): pp. 162-273.
Supplemental: PreCalculus: graphical, numerical, algebraic, Pearson (2004):
Resource Manual pp. 31-54
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 19 of 71
Columbus Public Schools 7/20/05
Polynomial, Power, Rational- A
Patterns in Polynomial End Behavior
Name
The following polynomial functions all have a few things in common. First, see if you can
determine how they are algebraically similar:
f ( x) = 2 x 2
f ( x) = 7 x 4 − x 3 + 6 x 2 + 11
f ( x) = 9 x10 − 3 x 7
f ( x) = 0.00032 x18 − 5
What do these functions have in common algebraically?
Next, try graphing 2 or more of these.
How are the graphs similar?
Fill out the 1st row of the summary chart at the end of this exercise.
This next set is related to the previous set, but there is one crucial difference. See if you can
identify the difference.
f ( x ) = −2 x 2
f ( x) = −7 x 4 − x 3 + 6 x 2 + 11
f ( x) = −9 x10 − 3 x 7
f ( x) = −0.00032 x18 − 5
How are these functions algebraically similar to each other (and different from the first set?).
You may want to revise your answers to the 1st set.
How are the graphs similar (and different from the first set?)
Fill out the 2nd row of the summary chart at the end of this exercise.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 20 of 71
Columbus Public Schools 7/20/05
Considering the answers to the first two sets, what makes these different?
Polynomial, Power, Rational- A
f ( x) = 7 x
f ( x) = x 3 − 4 x
f ( x) = 0.004 x 7 + x 4 − 5
f ( x) = x 5 − 2 x 4 + 3 x 3 + 4 x 2 + 5 x + 1
How are these functions algebraically similar to each other(and different from the first two
sets?).
How are the graphs similar to each other (and different from the first two sets?)
Fill out the 3rd row of the summary chart at the end of this exercise.
Now for the last set:
f ( x ) = −7 x
f ( x) = − x3 − 4 x
f ( x) = −0.004 x 7 + x 4 − 5
f ( x) = − x5 − 2 x 4 + 3 x3 + 4 x 2 + 5 x + 1
How are these functions algebraically similar to each other (and different from the first two
sets?)?
How are the graphs similar to each other (and different from the first two sets)?
Fill out the 4th row of the summary chart at the end of this exercise.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 21 of 71
Columbus Public Schools 7/20/05
Summary Chart
Leading Term
properties
Polynomial, Power, Rational- A
Sketch of Generic Graph
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
lim
x→−∞
Page 22 of 71
lim
x→∞
Columbus Public Schools 7/20/05
Polynomial, Power, Rational- A
Patterns in Polynomial End Behavior
Answer Key
The following polynomial functions all have a few things in common. First, see if you can
determine how they are algebraically similar:
f ( x) = 2 x 2
f ( x) = 7 x 4 − x 3 + 6 x 2 + 11
f ( x) = 9 x10 − 3 x 7
f ( x) = 0.00032 x18 − 5
What do these functions have in common algebraically?
They have even-powered leading terms with a positive coefficient.
Next, try graphing 2 or more of these.
How are the graphs similar?
They go up at both ends.
Fill out the 1st row of the summary chart at the end of this exercise.
This next set is related to the previous set, but there is one crucial difference. See if you can
identify the difference.
f ( x ) = −2 x 2
f ( x) = −7 x 4 − x 3 + 6 x 2 + 11
f ( x) = −9 x10 − 3 x 7
f ( x) = −0.00032 x18 − 5
How are these functions algebraically similar to each other (and different from the first set?).
You may want to revise your answers to the 1st set.
They have even-powered leading terms with a negative coefficient.
How are the graphs similar (and different from the first set?)
They go downwards at both ends.
Fill out the 2nd row of the summary chart at the end of this exercise.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 23 of 71
Columbus Public Schools 7/20/05
Considering the answers to the first two sets, what makes these different?
Polynomial, Power, Rational- A
f ( x) = 7 x
f ( x) = x 3 − 4 x
f ( x) = 0.004 x 7 + x 4 − 5
f ( x) = x 5 − 2 x 4 + 3 x 3 + 4 x 2 + 5 x + 1
How are these functions algebraically similar to each other (and different from the first two
sets)?
They have odd-powered leading terms with a positive coefficient.
How are the graphs similar to each other (and different from the first two sets)?
They go downwards on the left side, and upwards on the right side.
Fill out the 3rd row of the summary chart at the end of this exercise.
Now for the last set:
f ( x ) = −7 x
f ( x) = − x3 − 4 x
f ( x) = −0.004 x 7 + x 4 − 5
f ( x) = − x5 − 2 x 4 + 3 x3 + 4 x 2 + 5 x + 1
How are these functions algebraically similar to each other (and different from the first two
sets)?
They have odd-powered leading terms with a negative coefficient.
How are the graphs similar to each other (and different from the first two sets)?
They go upwards on the left side, and downwards on the right side.
Fill out the 4th row of the summary chart at the end of this exercise.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 24 of 71
Columbus Public Schools 7/20/05
Summary Chart
Leading Term
properties
Polynomial, Power, Rational- A
Sketch of Generic Graph
lim
lim
x→−∞
x→∞
Even-powered,
positive
coefficient
∞
∞
Even-powered,
negative
coefficient
-∞
-∞
-∞
∞
∞
-∞
Odd-powered,
positive
coefficient
Odd-powered,
negative
coefficient
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 25 of 71
Columbus Public Schools 7/20/05
Home Run
Polynomial, Power, Rational- A
Name
In 1919, Babe Ruth hit what some experts called the longest home run ever recorded in major
league baseball. In an exhibition game between the Boston Red Sox and the New York Giants,
he sent the ball along a parabolic trajectory. The trajectory of the ball is given by the equation.
y = x - .0017x2, where x represents the horizontal distance in feet and y is the vertical distance in
feet of the ball from home plate.
Find the answers to the questions using algebra and graphically. Show your work.
What was the greatest height reached by the ball?
How far from home plate did the ball land?
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 26 of 71
Columbus Public Schools 7/20/05
Home Run
Answer Key
Polynomial, Power, Rational- A
In 1919, Babe Ruth hit what some experts called the longest home run ever recorded in major
league baseball. In an exhibition game between the Boston Red Sox and the New York Giants,
he sent the ball along a parabolic trajectory. The trajectory of the ball is given by the equation.
y = x - .0017x2, where x represents the horizontal distance in feet and y is the vertical distance in
feet of the ball from home plate.
Find the answers to the questions using algebra and graphically. Show your work.
What was the greatest height reached by the ball?
x=
−1 = 294.118 f(294.118) = 147.059 The maximum height was 147.059 feet.
−.0034
How far from home plate did the ball land?
x - .0017 x2 = 0
x(1 - .0017x) = 0
x = 0 or x = 588.235. The ball landed 588.235 feet from home plate.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 27 of 71
Columbus Public Schools 7/20/05
Graphs, Factors, Zeros
Polynomial, Power, Rational- A
Name
Fill in the missing parts of the chart. The first two sections are done as examples. Look for the
relationship between the x-intercepts, the factors, and the zeros.
Graph, x-intercepts
y = 3 x+ 6
Factor
3x + 6
3(x + 2)
Zeros
3x + 6 = 0
3(x + 2) = 0
x = -2
y = x2 + 2x − 8
x2 + 2x − 8
(x + 4)(x − 2)
x2 + 2x − 8 = 0
(x + 4)(x − 2) = 0
x = -4, x = 2
y = x2 − 2x − 8
x2 − 2x − 8
x2 − 2x − 8 = 0
y = x3+ x2 − 4x 4
x3+ x2 − 4x − 4
x3 + x2 − 4x − 4 = 0
What is the relationship between the x-intercepts, the factors, and the zeros?
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 28 of 71
Columbus Public Schools 7/20/05
Graphs, Factors, Zeros
Answer Key
Polynomial, Power, Rational- A
Fill in the missing parts of the chart. The first two sections are done as examples. Look for the
relationship between the x-intercepts, the factors, and the zeros.
Graph, x-intercepts
y = 3x + 6
Factor
3x + 6
Zeros
3x + 6 = 0
3(x + 2)
3(x + 2) = 0
x = -2
y = x2+2x − 8
y = x2 − 2x − 8
x2 + 2x − 8
(x + 4)(x - 2)
x2 + 2x − 8 = 0
(x + 4)(x − 2) = 0
x = -4, x = 2
x2 − 2x − 8
x2 − 2x − 8 = 0
(x – 4)(x + 2)
(x – 4)(x + 2)=0
x = 4, x = -2
y = x3+ x2 − 4x − 4
x3+ x2 − 4 x − 4
x3+ x2 − 4x - 4 = 0
(x + 2) (x + 1)(x − 2)
(x + 2) (x + 1)(x − 2) = 0
x = -2, x = -1, x = 2
What is the relationship between the x-intercepts, the factors, and the zeros?
The x-intercepts are the zeros of the function and the solution to the equation. The factors
are of the form (x - a) where a is a zero.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 29 of 71
Columbus Public Schools 7/20/05
Connections with Multiplicities
Polynomial, Power, Rational- A
Name
When a polynomial is expressed in factored form, it is easy to see the multiplicity of each zero.
For example in the polynomial, P(x) = (x – 2)3 (x – 1)2 (x + 2), the multiplicity of the first zero
(2) is 3, the multiplicity of the second zero (1) is 2, and the multiplicity of the third zero (-2) is 1.
The multiplicity of each zero causes a definite behavior in the graph. Look at the graph of P(x)
on various viewing windows. The first graph shows all the zeros (if you look carefully) and is a
complete graph. The second graph zooms in on the zeros at 1 and 2. Notice that the graph
crosses the x-axis at -2 and 2, but just touches it at 1. Also notice the change in concavity
between 1 and 2.
[-5,5] by [-150, 10]
[.5, 2.5] by [-.5,.5]
In this exercise, you will graph several polynomials. You will use your calculator to make a
graph so that you can see all the zeros and the behavior near the zeros. You do not need to
sketch a complete graph. Sketch the graphs below each polynomial. When you are finished
graphing, complete the chart and the questions on the next page.
1. P(x) = (x – 2)4 (x – 1)2 (x + 2)
2. P(x) = (x – 2)5 (x – 1)3 (x + 2)
3. P(x) = (x – 2)4 (x – 1)2 (x + 2)2
4. P(x) = (x – 2)3 (x – 1)2 (x + 2)
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 30 of 71
Columbus Public Schools 7/20/05
Polynomial, Power, Rational- A
On the chart, for each zero of the polynomial, give the multiplicity, tell whether the multiplicity
is odd or even, and give the number of changes in concavity between this zero and the previous
one.
Polynomial
Zero
P(x) = (x – 2)4 (x – 1)2 (x + 2)
2
Multiplicity
Odd or Even
Multiplicity
Crosses or
Touches x-axis
1
-2
P(x) = (x – 2)5 (x – 1)3 (x + 2)
2
1
-2
P(x) = (x – 2)4 (x – 1)2 (x + 2)2
2
1
-2
P(x) = (x – 2)3 (x – 1)2 (x + 2)
2
1
-2
How can you tell from the factored equation whether the graph will touch or cross the x-axis?
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 31 of 71
Columbus Public Schools 7/20/05
Polynomial, Power, Rational- A
Connections with Multiplicities
Answer Key
When a polynomial is expressed in factored form, it is easy to see the multiplicity of each zero.
For example in the polynomial, P(x) = (x – 2)3 (x – 1)2 (x + 2), the multiplicity of the first zero
(2) is 3, the multiplicity of the second zero (1) is 2, and the multiplicity of the third zero ( -2) is
1. The multiplicity of each zero causes a definite behavior in the graph. Look at the graph of
P(x) on various viewing windows. The first graph shows all the zeros (if you look carefully) and
is a complete graph. The second graph zooms in on the zeros at 1 and 2. Notice that the graph
crosses the x-axis at -2 and 2, but just touches it at 1. Also notice the change in concavity
between 1 and 2.
[-5,5] by [-150, 10]
[.5, 2.5] by [-.5,.5]
In this exercise, you will graph several polynomials. You will use your calculator to make a
graph so that you can see all the zeros and the behavior near the zeros. You do not need to
sketch a complete graph. Sketch the graphs below each polynomial. When you are finished
graphing, complete the chart and the questions on the next page.
1. P(x) = (x – 2)4 (x – 1)2 (x + 2)
2. P(x) = (x – 2)5 (x – 1)3 (x + 2)
3. P(x) = (x – 2)4 (x – 1)2 (x + 2)2
4. P(x) = (x – 2)3 (x – 1)2 (x + 2)1
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 32 of 71
Columbus Public Schools 7/20/05
Polynomial, Power, Rational- A
On the chart, for each zero of the polynomial, give the multiplicity, tell whether the multiplicity
is odd or even and whether the graph touch or cross the x-axis.
Polynomial
Zero
Multiplicity
Odd or Even
Multiplicity
Crosses or
Touches x-axis
P(x) = (x – 2)4 (x – 1)2 (x + 2)
2
4
Even
Touches
1
2
Even
Touches
-2
1
Odd
Crosses
2
5
Odd
Crosses
1
3
Odd
Crosses
-2
1
Odd
Crosses
2
4
Even
Touches
1
2
Even
Touches
-2
2
Even
Touches
2
3
Odd
Crosses
1
2
Even
Touches
-2
1
Odd
Crosses
P(x) = (x – 2)5 (x – 1)3 (x + 2)
P(x) = (x – 2)4 (x – 1)2 (x + 2)2
P(x) = (x – 2)3 (x – 1)2 (x + 2)
How can you tell from the factored equation whether the graph will touch or cross the x-axis?
If the multiplicity is even it touches the x-axis. If the multiplicity is odd it crosses the x-axis.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 33 of 71
Columbus Public Schools 7/20/05
Factors, Remainders, Zeros
Polynomial, Power, Rational- A
Name
1. Multiply (2x - 3)(3x + 2)(5x - 1)
2. Explain which numbers are multiplied together to give the first coefficient of the answer in
Problem 1.
Which are multiplied together to give the constant term of the answer in Problem 1?
3. Give the zeros of (2x - 3)(3x + 2)(5x - 1) = 0
4. Notice how the zeros are related to the first and last coefficients of the polynomial in
Problem 1
The polynomial above was 30 x3- 31 x2 - 25 x + 6. Therefore the zeros of
f(x) = 30x3- 31x2 - 25x + 6 are 3/2, -2/3, and 1/5. The 3, the -2, and the 1 in the numerator came
from the opposite of the second term in each binomial of the original product. The 4, the 3, and
the 5 in the denominator came from the first term in each binomial in the original product.
Without looking at the factored form of the polynomial, the zeros of f(x) = 30 x3- 31 x2- 25 x + 6
are not immediately obvious. But it is possible to narrow down any possible zeros of the form
r/s (rational zeros) because any values of r must be factors of 6 and any values of s must be
factors of 30.
In general, this is the Rational Zero Theorem. For a polynomial function
P(x) = an xn + an-1 xn-1 + an-2 xn-2 +.…+ a1 x1 + a0, if there are rational zeros, they must be of the form
r/s where values of r must be factors of a0 and any values of s must be factors of an.
Combining this with the Factor Theorem, the graphing calculator, and synthetic division
provides a method for factoring polynomials and finding some complex zeros.
5. List all possible rational zeros of P(x) = 10 x4-17 x3 + 43 x2 -68 x+12.
6. Use your graphing calculator to graph the function. Find the zeros graphically and determine
which of them (if any) correspond to rational zeros by testing them with the Factor Theorem.
(If r/s is a factor, then P(r/s) = 0.)
7. Select one of the zeros (call it d) and use synthetic division to divide the polynomial by
(x – d).
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 34 of 71
Columbus Public Schools 7/20/05
Polynomial, Power, Rational- A
8. Take the polynomial that is the dividend and divide it by the other rational zero.
9. The remaining polynomial has no real zeros. Use the quadratic formula to find the nonreal
zeros.
10. Use this method to find any nonreal zeros of P(x) = 60 x5- 221 x4 - 48 x3 - 145 x2 + 114 x + 72
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 35 of 71
Columbus Public Schools 7/20/05
Factors, Remainders, Zeros
Answer Key
Polynomial, Power, Rational- A
Multiply (2 x - 3)(3 x+ 2)(5 x- 1) 30 x3 - 31 x2 - 25 x + 6
1. Explain which numbers are multiplied together to give the first coefficient of the answer in
Problem 1. 2, 3, and 5
2. Which are multiplied together to give the constant term of the answer in Problem 1? -3, 2, -1
3. Give the zeros of (2 x - 3)(3 x+ 2)(5 x- 1) = 0 3/2, -2/3, 1/5
4. Notice how the zeros are related to the first and last coefficients of the polynomial in
Problem 1.
The polynomial above was 30 x3- 31 x2 - 25 x + 6. Therefore the zeros of
f(x) = 30x3- 31x2 - 25x + 6 are 3/2, -2/3, and 1/5. The 3, the -2, and the 1 in the numerator came
from the opposite of the second term in each binomial of the original product. The 4, the 3, and
the 5 in the denominator came from the first term in each binomial in the original product.
Without looking at the factored form of the polynomial, the zeros of f(x) = 30 x3- 31 x2- 25 x + 6
are not immediately obvious. But it is possible to narrow down any possible zeros of the form
r/s (rational zeros) because any values of r must be factors of 6 and any values of s must be
factors of 30.
In general, this is the Rational Zero Theorem. For a polynomial function
P(x) = an xn + an-1 xn-1 + an-2 xn-2 +.…+ a1 x1 + a0, if there are rational zeros, they must be of the form
r/s where values of r must be factors of a0 and any values of s must be factors of an.
Combining this with the Factor Theorem , the graphing calculator, and synthetic division
provides a method for factoring polynomials and finding some complex zeros.
5. List all possible rational zeros of P(x) = 10 x4-17 x3 + 43 x2 -68 x+12
1, 2, 3, 4, 6,12
±
1, 2, 5,10
6. Use your graphing calculator to graph the function. Find the zeros graphically and determine
which of them (if any) correspond to rational zeros by testing them with the Factor Theorem.
(If r/s is a factor, then P(r/s) = 0.)
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 36 of 71
Columbus Public Schools 7/20/05
Polynomial, Power, Rational- A
7. Select one of the zeros (call it d) and use synthetic division to divide the polynomial by
(x – d). 5x3 - x2 + 20x – 4
8. Take the polynomial that is the divisor and divide it by the other rational zero.
x2+4
9. The remaining polynomial has no real zeros. Use the quadratic formula to find the nonreal
zeros. x = ±2i
.
10. Use this method to find all zeros of P(x) = 60 x5- 221 x4 - 48 x3 - 145 x2 + 114 x + 72
4, -2/5, 3/4, −1 ± 2i 3
3
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 37 of 71
Columbus Public Schools 7/20/05
Exponential vs. Power
Polynomial, Power, Rational- B
Name
Graph y = x2 and y = 2x in the Decimal (ZOOM4) window. Sketch your graph below. Which
graph appears to be increasing the fastest? How many solutions do you think there are to the
equation x2 = 2x?
Graph the same functions in the Standard (ZOOM6) window. Sketch your graph below. Which
graph appears to be increasing the fastest? How many solutions do you think there are to the
equation x2 = 2x?
Graph the same functions in the window [0,10] by [0,20]. Sketch your graph below. Which
graph appears to be increasing the fastest? How many solutions do you think there are to the
equation x2 = 2x? Find all the solutions to the equation.
Graph y = x4 and y = 4x in the Decimal (ZOOM4) window. Sketch your graph below. Which
graph appears to be increasing the fastest? How many solutions do you think there are to the
equation x4 = 4x?
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 38 of 71
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Polynomial, Power, Rational- B
Graph y = x4 and y = 4x in the Standard (ZOOM6) window. Sketch your graph below. Which
graph appears to be increasing the fastest? How many solutions do you think there are to the
equation x4 = 4x?
Graph the same functions in the window [0, 3] by [0, 20]. Sketch your graph below. Which
graph appears to be increasing the fastest? How many solutions do you think there are to the
equation x4 = 4x?
Graph the same functions in the window [3.75, 240] by [4.25, 275]. Sketch your graph below.
Which graph appears to be increasing the fastest? How many solutions do you think there are to
the equation x4 = 4x? Find all the solutions to the equation.
Consider the equation x8=2x. How many solutions do you think it has? Graph in several
windows. Do you think you can find all the solutions?
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 39 of 71
Columbus Public Schools 7/20/05
Exponential vs. Power
Answer Key
Polynomial, Power, Rational- B
Graph y = x2 and y = 2x in the Decimal (ZOOM4) window. Sketch your graph below. Which
graph appears to be increasing the fastest? How many solutions do you think there are to the
equation x2 = 2x? Answers will vary.
Graph the same functions in the Standard (ZOOM6) window. Sketch your graph below. Which
graph appears to be increasing the fastest? How many solutions do you think there are to the
equation x2 = 2x? Answers will vary.
Graph the same functions in the window [0,10] by [0,20]. Sketch your graph below. Which
graph appears to be increasing the fastest? How many solutions do you think there are to the
equation x2 = 2x? Find all the solutions to the equation. x = 4 or x = 2 or x = -.766665
Graph y = x4 and y = 4x in the Decimal (ZOOM4) window. Sketch your graph below. Which
graph appears to be increasing the fastest? How many solutions do you think there are to the
equation x4 = 4x? Answers will vary.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 40 of 71
Columbus Public Schools 7/20/05
Polynomial, Power, Rational- B
Graph y = x4 and y = 4x in the Standard (ZOOM6) window. Sketch your graph below. Which
graph appears to be increasing the fastest? How many solutions do you think there are to the
equation x4 = 4x? Answers will vary.
Graph the same functions in the window [0,3] by [0,20]. Sketch your graph below. Which
graph appears to be increasing the fastest? How many solutions do you think there are to the
equation x4 = 4x? Answers will vary.
Graph the same functions in the window [3.75,240] by [4.25,275]. Sketch your graph below.
Which graph appears to be increasing the fastest? How many solutions do you think there are to
the equation x4 = 4x? Find all the solutions to the equation. x = 4. or x = 2. or x = -.766665
Consider the equation x8=2x. How many solutions do you think it has? Graph in several
windows. Do you think you can find all the solutions? Answers will vary. The solutions are
x = 43.5593 or x =1.1 or x = -.923132
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 41 of 71
Columbus Public Schools 7/20/05
Investigating Pendulum Length
Polynomial, Power, Rational- B
Name
The pendulum of a clock swings back and forth, causing the clock to tick off the seconds. A
pendulum clock runs down over time. With each swing, the pendulum travels a slightly smaller
distance. However, even as the clock starts to run down, it can still keep good time. The reason
is because the period of a pendulum, or the amount of time it takes the pendulum to swing from
point A to point B and back to point A, is the same for large swings and small swings. Your
group will investigate how the length of a pendulum affects the period.
Small
swing
A
Materials:
B
Large
swing
A
B
4 pennies, a stop watch, tape, a pencil, a ruler, strings
Procedure:
- To make your pendulum, tape 4 pennies to a piece of string.
- Use 10 different lengths of string that range between 10 cm and 30 cm. Record that
in the table below.
- Use a pencil to hold the pendulum away from the table.
-
Determine how many seconds it takes your pendulum to make 10 full swings. Divide
by 10 to find the period. Record the period in the table below for each string length.
Data Collection:
String Length (cm)
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Period (sec)
Page 42 of 71
Columbus Public Schools 7/20/05
Polynomial, Power, Rational- B
Analysis:
1. Which is the dependent variable/quantity and which is the independent variable/quantity?
2. Enter your data into your calculator and have the calculator make a scatter plot. Sketch the
scatter plot below and label your axes.
3. Based on your collected data, estimate a string length that produces a period of one second.
Can you keep time with this pendulum?
4. Use your calculator and find the Power Regression Equation ( y
and round to three decimal places.
= ax b ). Record it below
5. Using your equation in #4, find the string length that produces a period of one second. Show
algebraic work. Check your answer by solving graphically. Round answer to two decimal
places. How does this answer compare to your estimate in #3?
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 43 of 71
Columbus Public Schools 7/20/05
Polynomial, Power, Rational- B
Extension:
The Italian scientist Galileo Galilei (1564-1642) showed that the period of a pendulum is
proportional to the square root of the length of the pendulum. He discovered the formula that
relates the period, P, of a pendulum to its length, L, is
is in centimeters.
P = 0.2 L where P is in seconds and L
1. How does your regression equation compare to the equation P = 0.2 L ? Graph the
equation P = 0.2 L along with your scatter plot and your regression equation on your
graphing calculator. How do they compare? Be specific.
2. Use the equation P = 0.2 L to find the length L of a pendulum that has a period of one
second. Show algebraic work. Check your answer by solving graphically. Round to two
decimal places. How does this answer compare to your answer in #5 of Analysis?
3. The first pendulum clock was invented in 1657. How does a pendulum clock work?
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 44 of 71
Columbus Public Schools 7/20/05
Investigating Pendulum Length
Answer Key
Polynomial, Power, Rational- B
The pendulum of a clock swings back and forth, causing the clock to tick off the seconds. A
pendulum clock runs down over time. With each swing, the pendulum travels a slightly smaller
distance. However, even as the clock starts to run down, it can still keep good time. The reason
is because the period of a pendulum, or the amount of time it takes the pendulum to swing from
point A to point B and back to point A, is the same for large swings and small swings. Your
group will investigate how the length of a pendulum affects the period.
Small
swing
A
Materials:
B
Large
swing
A
B
4 pennies, a stop watch, tape, a pencil, a ruler, strings
Procedure:
- To make your pendulum, tape 4 pennies to a piece of string.
- Use 10 different lengths of string that range between 10 cm and 30 cm. Record that
in the table below.
- Use a pencil to hold the pendulum away from the table.
-
Determine how many seconds it takes your pendulum to make 10 full swings. Divide
by 10 to find the period. Record the period in the table below for each string length.
Data Collection: Sample data below.
String Length (cm)
Period (sec)
10
12
14
16
20
22
23
25
28
30
0.6
0.7
0.75
0.8
0.9
0.94
0.96
1
1.05
1.1
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 45 of 71
Columbus Public Schools 7/20/05
Polynomial, Power, Rational- B
Analysis:
1. Which is the dependent variable/quantity and which is the independent variable/quantity?
Period is dependent; length is independent.
2. Enter your data into your calculator and have the calculator make a scatter plot. Sketch the
scatter plot below and label your axes. Answers will vary
3. Based on your collected data, estimate a string length that produces a period of one second.
Can you keep time with this pendulum? About 25 cm. Only for a short period of time.
The pendulum stops swinging before much time has elapsed.
4. Use your calculator and find the Power Regression Equation ( y = ax ). Record it below
and round to three decimal places. Using sample data above, y= .187x.522
b
5. Using your equation in #4, find the string length that produces a period of one second. Show
algebraic work. Check your answer by solving graphically. Round answer to two decimal
places. How does this answer compare to your estimate in #3? 24.87 cm.
Answers will vary
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 46 of 71
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Polynomial, Power, Rational- B
Extension:
The Italian scientist Galileo Galilei (1564-1642) showed that the period of a pendulum is
proportional to the square root of the length of the pendulum. He discovered the formula that
relates the period, P, of a pendulum to its length, L, is
is in centimeters.
P = 0.2 L where P is in seconds and L
1. How does your regression equation compare to the equation P = 0.2 L ? Graph the
equation P = 0.2 L along with your scatter plot and your regression equation on your
graphing calculator. How do they compare? Be specific. Answers will vary
2. Use the equation P = 0.2 L to find the length L of a pendulum that has a period of one
second. Show algebraic work. Check your answer by solving graphically. Round to 2
decimal places. How does this answer compare to your answer in #5 of Analysis? 25 cm.
Answers will vary
3. The first pendulum clock was invented in 1657. How does a pendulum clock work? Because
the period of the clock is determined only by the length of the pendulum and gravity,
the length is chosen so that the pendulum swings every second or two seconds, twice a
second, etc. depending on the size of the clock. On some clocks the length of the
pendulum can be adjusted slightly to account for individual differences. A system of
weights and gears or a winding mechanism and gears allows the pendulum to swing for
longer periods of time.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 47 of 71
Columbus Public Schools 7/20/05
Rational Functions
Polynomial, Power, Rational- C
Name
Rational Functions: a function f(x) is a rational function if and only if it can be expressed as the
g ( x)
result of dividing two polynomial functions, i.e. f ( x) =
where both g(x) and h(x) are
h( x )
polynomial functions in the same variable.
Which of the following are rational functions?
A. ν 4 − 5ν 2 + 7
H. (h + 5)(h + 4)(h + 3)(h + 2)(h + 1)
B.
g 3 + 2 g 2 − 8g
g +1
I.
C.
x2 − 2 x + 8
x−6
J. 7 x + 5 −
D.
y +3
y
1
c + c 3 − 2c 2
5
x −8
x2
3a
2
K. 2a + 1
5a + 3
2a 2 + 1
k 4 + 2k 3 − 7 k 2 + 11
E.
(k + 3)(k − 8)
3 j2 − 4
L.
7m + 2
(r + 6)(r − 3) 2
F.
r (r + 1)(r + 2)
⎛ ( z − 3 )5 ⎞
M. ⎜
⎟
⎜ z+9 ⎟
⎝
⎠
5− x
G.
3x 2
⎛ t +7 ⎞
N. ⎜ 3
⎟
⎝ 5t − 4 ⎠
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 48 of 71
4
−6
Columbus Public Schools 7/20/05
Rational Functions
Answer Key
Polynomial, Power, Rational- C
Rational Functions: a function f(x) is a rational function if and only if it can be expressed as the
g ( x)
where both g(x) and h(x) are
result of dividing two polynomial functions, i.e. f ( x) =
h( x )
polynomial functions in the same variable.
Which of the following are rational functions?
A. ν 4 − 5ν 2 + 7
H. (h + 5)(h + 4)(h + 3)(h + 2)(h + 1)
Not Rational
g 3 + 2 g 2 − 8g
B.
g +1
Not Rational
I.
Rational
x2 − 2 x + 8
C.
x−6
Rational
J. 7 x + 5 −
Not Rational
D.
y +3
y
Not Rational
E.
k 4 + 2k 3 − 7 k 2 + 11
(k + 3)(k − 8)
Rational
(r + 6)(r − 3) 2
F.
r (r + 1)(r + 2)
Rational
G.
5− x
3x 2
Not Rational
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
1
c + c 3 − 2c 2
5
x −8
x2
Rational
3a
2
K. 2a + 1
5a + 3
2a 2 + 1
Rational
L.
3 j2 − 4
7m + 2
Not Rational
⎛ ( z − 3 )5 ⎞
M. ⎜
⎟
⎜ z+9 ⎟
⎝
⎠
4
Rational
⎛ t +7 ⎞
N. ⎜ 3
⎟
⎝ 5t − 4 ⎠
−6
Rational
Page 49 of 71
Columbus Public Schools 7/20/05
Graphing Rational Functions
Polynomial, Power, Rational- C
Name
Graph each of the following functions on a set of coordinate axes. Be sure to indicate any
asymptotes by drawing them as dotted lines, and answer any questions:
Group 1
1
A) y =
x
B) y =
1
x−4
C) y =
1
+2
x
What are the similarities/differences among these graphs?
Group 2
( x + 4)
A) y =
( x − 5)
B) y =
( x + 4) 2
( x − 5)
C) y =
( x + 4)3
( x − 5)
How do these graphs differ with respect to what happens as x approaches negative infinity, and
as x approaches infinity?
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 50 of 71
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Group 3
4
A) y =
( x + 3)
4
B) y =
( x − 3)( x + 1)
Polynomial, Power, Rational- C
C) y =
4
( x − 3)( x + 1)( x − 6)
How can we determine the number (and location) of vertical asymptotes from the equation?
Group 4
A) y = 3 x
B) y =
3 x( x + 4)
( x + 4)
C) y =
3x( x + 4)( x − 2)
( x + 4)( x − 2)
This last set is extremely tricky. We see from these equations and graphs that although these
functions are very similar, there is in fact something different about them. Using your trace and
zoom features, try to determine what that difference is.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 51 of 71
Columbus Public Schools 7/20/05
Graphing Rational Functions
Answer Key
Polynomial, Power, Rational- C
Graph each of the following functions on a set of coordinate axes. Be sure to indicate any
asymptotes by drawing them as dotted lines, and answer any questions:
Group 1
1
A) y =
x
B) y =
1
x−4
C) y =
1
+2
x
What are the similarities/differences among these graphs?
They are all the same basic shape. B looks like A, after A has been shifted 4 units to the
right. C looks like A after A has been shifted 2 units up.
Group 2
( x + 4)
A) y =
( x − 5)
B) y =
( x + 4) 2
( x − 5)
C) y =
( x + 4)3
( x − 5)
How do these graphs differ with respect to what happens as x approaches negative infinity, and
as x approaches infinity?
In A, y appears to approach 0 at both ends. In B, y keeps decreasing on the negative x side,
and keeps increasing on the positive x side, approaching the same line. In C, y keeps
increasing on the negative x side, and keeps decreasing on the positive x side, not
approaching any line at either end.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 52 of 71
Columbus Public Schools 7/20/05
Group 3
4
A) y =
( x + 3)
4
B) y =
( x − 3)( x + 1)
Polynomial, Power, Rational- C
C) y =
4
( x − 3)( x + 1)( x − 6)
How can we determine the number (and location) of vertical asymptotes from the equation?
Each value of x that yields a denominator of zero will have a vertical asymptote. We can
find them by setting each individual factor equal to zero and then solving. For example, in
C we know (x - 3) = 0, (x + 1) = 0, and (x – 6 ) = 0, so the asymptotes are at x = 3, x = -1, and
x = 6.
Group 4
A) y = 3 x
B) y =
3 x( x + 4)
( x + 4)
C) y =
3x( x + 4)( x − 2)
( x + 4)( x − 2)
This last set is extremely tricky: We see from these equations and graphs that although these
functions are very similar, there is in fact something different about them. Using your trace and
zoom features, try to determine what that difference is.
The graphs appear identical, but in B we have one point dropped out (a removable
discontinuity) at (-4,-12), and in C we have two points dropped out (removable
discontinuities) at (-4,-12) and (2,6). The easiest way to see these by using TRACE is to
perform a ZOOM DECIMAL, and then trace along the “line” until we get x-values where
no y-value is given.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 53 of 71
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Polynomial, Power, Rational- C
End-Behavior of Rational Functions
Name
Graph each of the following rational functions on the axes provided:
Set 1:
3
y=
x
−2x 2
y=
11x 7
5x
y=
11x 2
x4
y= 5
x
What happens to y in each graph as x approaches positive infinity?
What happens to y in each graph as x approaches negative infinity?
Looking at only the degree of the numerator and of the denominator, what is similar about the
form of each function?
Set 2:
4x
y=
x+6
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
−15x 3
y= 3
5x + 2x − 6
Page 54 of 71
7x 8
y= 8
x − x7 + x6 − 3
Columbus Public Schools 7/20/05
What happens to y in each graph as x approaches positive infinity?
Polynomial, Power, Rational- C
What happens to y in each graph as x approaches negative infinity?
Looking at only the degree of the numerator and of the denominator, what is similar about the
form of each function? What is different? How can we predict the end-behavior?
Set 3:
x2 − 8
y=
x+5
−2x 3 + 7x 2 + 8
y=
x2 − 1
7x 4 − 1
y= 3
x +4
What happens to y in each graph as x approaches positive infinity?
What happens to y in each graph as x approaches negative infinity?
Looking at only the degree of the numerator and of the denominator, what is similar about the
form of each function? What is different? How can we predict the end-behavior?
Wrap-Up: What indicators can we use to predict the end behavior of rational functions?
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 55 of 71
Columbus Public Schools 7/20/05
Polynomial, Power, Rational- C
End-Behavior of Rational Functions
Answer Key
Graph each of the following rational functions on the axes provided:
Set 1:
3
y=
x
−2x 2
y=
11x 7
5x
y=
11x 2
x4
y= 5
x
What happens to y in each graph as x approaches positive infinity?
y approaches zero.
What happens to y in each graph as x approaches negative infinity?
y approaches zero.
Looking at only the degree of the numerator and of the denominator, what is similar about the
form of each function?
The degree of the numerator is less than the degree of the denominator
Set 2:
4x
y=
x+6
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
−15x 3
y= 3
5x + 2x − 6
Page 56 of 71
7x 8
y= 8
x − x7 + x6 − 3
Columbus Public Schools 7/20/05
What happens to y in each graph as x approaches positive infinity?
y approaches a horizontal line, the values 4, -3, and 7 respectively.
Polynomial, Power, Rational- C
What happens to y in each graph as x approaches negative infinity?
y approaches a horizontal line, the values 4, -3, and 7 respectively.
Looking at only the degree of the numerator and of the denominator, what is similar about the
form of each function? What is different? How can we predict the end-behavior?
The degrees of the numerator and denominator are the same. We can predict the
horizontal asymptote by dividing the leading coefficient of the numerator by the leading
coefficient of the denominator--this gives the y-value that the function approaches.
Set 3:
x2 − 8
y=
x+5
−2x 3 + 7x 2 + 8
y=
x2 − 1
7x 4 − 1
y= 3
x +4
What happens to y in each graph as x approaches positive infinity?
y also approaches infinity, (or negative infinity for the second example) and appears to
approach a line. We have an oblique asymptote.
What happens to y in each graph as x approaches negative infinity?
y also approaches infinity, (or negative infinity for the second example) and appears to
approach a line. We have an oblique asymptote.
Looking at only the degree of the numerator and of the denominator, what is similar about the
form of each function? What is different? How can we predict the end-behavior?
The degree of the numerator is one greater than the degree of the denominator. We can
find the oblique asymptote by performing polynomial division on each rational function
and ignoring the remainder.
Wrap-Up: What indicators can we use to predict the end behavior of rational functions?
The degree and coefficients of the numerator and denominator can predict the kind of end
behavior and the specifics, as described in the previous answers.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 57 of 71
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Sign Chart Analysis
Polynomial, Power, Rational- C
Name
Consider the following rational function: f ( x) =
( x − 3)
( x + 2)
To help us graph this, we need to know where this function gives positive y values, and
where it gives negative y values. Start by considering zeros (at x = 3) and where the function is
undefined (at x = -2) and placing these on a number line. Note that these two numbers divide our
number line into three segments. Above the tick mark, put a 0 (if it's a zero) or U (if it's
undefined). Then place each factor over on the left side.
We will look at these factors one at a time. For the (x - 3) factor, it will be negative for
the interval (−∞, −2) , so we put a negative sign there. It is also negative for the interval (-2, 3),
so we put a negative sign there. It is positive for the interval (3, ∞) , so we put a + there.
Now consider the (x + 2) factor. It is negative for the first interval only, and positive for
the second and third intervals, so we mark our chart accordingly.
Finally, we can put this information together. For x values less than -2, we get a negative
divided by a negative, which gives us a positive value. For x values between -2 and 3, we get a
negative divided by a positive, which gives us a negative value. And for x values greater than 3,
we get a positive divided by a positive, which gives us a positive value.
(x + 2)
–
(x - 3)
–
RESULT:
+
+
U
-2
–
–
+
0
3
+
+
Now we'll let you analyze a more involved rational function:
( x − 2)( x + 4) 2 ( x + 1)
f ( x) =
x( x − 5)
First, determine
1. zeros:
2. undefined:
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 58 of 71
Columbus Public Schools 7/20/05
Now fill out the chart below:
Polynomial, Power, Rational- C
(x - 2)
(x + 4)2
(x + 1)
x
(x - 5)
RESULT:
Practice: Create a sign-chart analysis for the following rational functions:
1. f ( x) =
( x + 3)( x − 8)( x + 5)
( x − 8)( x + 7) 2
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 59 of 71
Columbus Public Schools 7/20/05
Polynomial, Power, Rational- C
What happens to the sign of y around where x equals 8? Why do you think that is the case?
What happens to the sign of y around where x equals –7? Why do you think that is the case?
−( x + 3) 2 ( x + 5)
2. f ( x) =
( x + 7) 2
2
Knowing what you do about multiplicity of zeros and of asymptotes, why do you think you get
the results you do?
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 60 of 71
Columbus Public Schools 7/20/05
Sign Chart Analysis
Answer Key
Consider the following rational function: f ( x) =
Polynomial, Power, Rational- C
( x − 3)
( x + 2)
To help us graph this, we need to know where this function gives positive y values, and
where it gives negative y values. Start by considering zeros (at x = 3) and where the function is
undefined (at x = -2) and placing these on a number line. Note that these two numbers divide our
number line into three segments. Above the tick mark, put a 0 (if it's a zero) or U (if it's
undefined). Then place each factor over on the left side.
We will look at these factors one at a time. For the (x - 3) factor, it will be negative for
the interval (−∞, −2) , so we put a negative sign there. It is also negative for the interval (-2, 3),
so we put a negative sign there. It is positive for the interval (3, ∞) , so we put a + there.
Now consider the (x+2) factor. It is negative for the first interval only, and positive for
the second and third intervals, so we mark our chart accordingly.
Finally, we can put this information together. For x values less than -2, we get a negative
divided by a negative, which gives us a positive value. For x values between -2 and 3, we get a
negative divided by a positive, which gives us a negative value. And for x values greater than 3,
we get a positive divided by a positive, which gives us a positive value.
(x+2)
–
(x-3)
–
+
U
–
-2
RESULT:
+
+
0
+
3
–
+
Now we'll let you analyze a more involved rational function:
(x − 2)(x + 4)2 (x + 1)
f (x) =
x(x − 5)
First, determine
zeros:
2, -4, -1
undefined:
0, -5
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 61 of 71
Columbus Public Schools 7/20/05
Now fill out the chart below:
Polynomial, Power, Rational- C
(x - 2)
–
–
–
–
+
+
( x + 4) 2
+
+
+
+
+
+
(x + 1)
–
–
+
+
+
+
x
–
–
–
+
+
+
(x - 5)
–
–
–
–
–
+
RESULT:
0
0
U
0
U
-4
-1
0
2
5
+
+
–
+
–
+
Practice: Create a sign-chart analysis for the following rational functions:
1. f ( x) =
( x + 3)( x − 8)( x + 5)
( x − 8)( x + 7) 2
(x - 3)
–
–
–
+
+
(x - 8)
–
–
–
–
+
(x + 5)
–
–
+
+
+
( x + 7) 2
+
+
+
+
+
U
-7
RESULT:
+
+
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
0
0
U
-5
3
8
+
Page 62 of 71
+
+
Columbus Public Schools 7/20/05
Polynomial, Power, Rational- C
What happens to the sign of y around where x equals 8? Why do you think that is the case?
The sign stays the same on both sides of 8 (positive) because 8 is a removable discontinuity.
What happens to the sign of y around where x equals –7? Why do you think that is the case?
2
The sign stays the same on both sides of –7 because the term (x − 7) always evaluates to
positive.
−( x + 3) 2 ( x + 5)
2. f ( x) =
( x + 7) 2
2
(x + 3)
+
+
+
+
( x + 5) 2
+
+
+
+
( x + 7) 2
+
+
+
+
U
-7
RESULT:
–
0
0
-5
-3
–
–
–
Knowing what you do about multiplicity of zeros and of asymptotes, why do you think you get
the results you do?
Both zeros are even, so the graph only touches the axis at both of those x-intercepts (–3 and
–5), staying negative. The vertical asymptote, x = –7 is also even, so the graph doesn't
switch from positive to negative or negative to positive at that point.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 63 of 71
Columbus Public Schools 7/20/05
Keep on Bouncing
Teacher Notes - D
This activity is designed to demonstrate modeling quadratics, practice the use of the Calculator
Based Ranger (CBR) and graphing calculator, and emphasize the need for class teamwork. The
follow-up activity described here is the teamwork payoff.
The activity requires that students work together in groups to collect data from a bouncing ball.
A soccer or playground ball is a good size, although it is possible to use a tennis ball or
racquetball. Students will probably need to collect data several times before they get good data.
If they get a set that they think looks marginal, they might want to switch calculators to run the
next trial. A sample data set looks like this.
The data for this is listed below. It is also available at this website. ftp://ftp.ti.com/pub/graphti/calc-apps/83/hsmotion/ and choose balldata.3p. The Graphing Calculator Resource Manual
PreCalculus Supplement describes how to download and install this data. It is a good idea to
have the data as a backup.
Time
0
0.041984
0.083968
0.125952
0.167936
0.20992
0.251904
0.293888
0.335872
0.377856
0.41984
0.461824
0.503808
0.545792
0.587776
0.62976
0.671744
Height
0.16712
0.7108
1.2207
1.65493
2.0333
2.35852
2.62699
2.845
2.9986
3.09545
3.14004
3.13013
3.06617
2.94635
2.76302
2.53149
2.24231
Time
0.713728
0.755712
0.797696
0.83968
0.881664
0.923648
0.965626
1.007606
1.049586
1.091566
1.133546
1.175526
1.217506
1.259486
1.301466
1.343446
1.385426
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Height
1.90312
1.50223
1.03737
0.49324
0
0.49774
0.95269
1.34683
1.6734
1.94186
2.15312
2.31078
2.41168
2.45672
2.44861
2.38375
2.26348
Time
1.427406
1.469386
1.511366
1.553346
1.595326
1.637306
1.679286
1.721266
1.763246
1.805226
1.847206
1.889186
1.931166
1.973146
2.015126
2.057106
2.099086
Page 64 of 71
Height
2.08691
1.85898
1.56935
1.2207
0.79323
0.34414
0.16982
0.58468
0.96215
1.27341
1.52971
1.72115
1.85718
1.93421
1.95853
1.927
1.83511
Time
2.141066
2.183046
2.225026
2.267006
2.308986
2.350966
2.392946
2.434926
2.476906
2.518886
2.560866
2.602846
2.644826
2.686806
2.728786
2.770766
2.812746
Height
1.69052
1.48286
1.22701
0.9108
0.53648
0.12748
0.30045
0.63648
0.92656
1.17251
1.35403
1.48061
1.55268
1.56214
1.5198
1.42701
1.26665
Columbus Public Schools 7/20/05
Time
2.854726
2.896706
2.938686
2.980666
3.022646
3.064626
3.106606
3.148586
3.190566
3.232546
3.274526
3.316506
3.358486
3.400466
3.442446
3.484426
3.526406
Height
1.05449
0.77612
0.44955
0.08063
0.32477
0.59729
0.84143
1.02611
1.1617
1.23242
1.24908
1.21124
1.1171
0.96305
0.7581
0.5018
0.19505
Time
3.568386
3.610366
3.652346
3.694326
3.736306
3.778286
3.820266
3.862246
3.904226
Height
0.16306
0.4572
0.65855
0.82882
0.94368
1.00539
1.01891
0.96035
0.85134
Teacher Notes - D
Before you begin the activity, instruct the students to show you their graphs before they begin to
analyze it. You should choose a good set of data and link it (Link lists L1 through L6) to the
teacher calculator and save L1 as BTIME and L2 as BDIST. You will need them in Topic 3.
(See the Graphing Calculator Resource manual if you are not sure how to do this.). Also have
one student in each group save the group’s complete data set as GTIME and GDIST.
It is necessary to save the data because the program used to select a parabola will delete all the
rest of the data. Also, if the data is left in L1 and L2, it will probably be overwritten before it is
needed in Topic 3.
After the students have completed the activity, distribute the data you collected and saved from a
student earlier (or use the sample data above) to the class, using Navigator, if available, or by
linking with students and having them link with each other. Assign each student or group of
students one of the parabolas to analyze. When they are finished, have students with the same
parabola agree on one equation, and then enter each of the equations into the teacher calculator
to see how well the equations match the scatterplot of the data. Also, if the equations are good,
the coefficient of the square term should be approximately -16 if using feet or approximately -4.9
if using meters because the acceleration of gravity in the traditional system of measure is -32
ft/sec2 or -9.8 m/ sec2 and the equation for the height of a falling body with respect to time is
d= ½ gt2 + vot + so.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 65 of 71
Columbus Public Schools 7/20/05
Keep on Bouncing
Polynomial, Power, Rational- D
Name
Materials Needed:
CBR (Ranger)
TI-82,\TI-83, or TI-84 Calculator
Link Cable
Ball
INSTRUCTIONS:
A. Setting up the calculator and Ranger
1. If you are using a TI-83 Plus, skip steps 1 and 2 and go to step 3.
If you are using a TI-82 or TI-83: Select PRGM on the keypad. If the program RANGER is
on the list select it and go to step 4.
2. If RANGER is not on the list, connect your calculator to the Ranger. On the calculator,
select 2nd Link. (It’s on the X key). Use the right arrow to highlight RECEIVE and hit
ENTER. The calculator will display Waiting… Open the RANGER and push the key 82/83.
The calculator should display DONE. Go to step 4.
3. On the TI-83 Plus or TI-84, choose APPS and choose CBL/CBR. (If it is not on the list,
follow the instructions for the 82 or 83.) Press any key. On the next screen select RANGER.
Go the step 4.
4. Hit ENTER. Select #3 Applications. When prompted for UNITS, select #2 FEET. Choose
#3, Ball Bounce.
B. Ball Bounce.
1. Be sure that the ball is dropped on a smooth, level surface. Do not allow anything to obstruct
the path between the Ranger and the ball while the data is being collected.
2. Follow the instructions on the calculator.
3. Your data should look like a series of parabolas, decreasing in height. Decide if you want to
try again or not.
4. Hit ENTER. If you did not like your graph, select #5, REPEAT SAMPLE and go back to
step #2. If you like your graph, go to step 5.
5. Choose #4, PLOT TOOLS. On PLOT TOOLS choose #1, SELECT DOMAIN. Pick out
your best parabola. For LEFT BOUND? use the right or left arrow to move the cursor to the
lowest point on the left side of the parabola you chose. Hit ENTER. For RIGHT BOUND,
use the right arrow to move to the lowest point on the right side of your parabola. Hit
ENTER.
Analysis
1. On your graph, what is measured on the x-axis?
the y-axis?
What is measured on
2. The ball was bouncing straight up and down. Why is the graph a series of parabolas? What
force makes the ball fall after each bounce? Why do the heights of the bounces decrease for
each bounce?
3. Use TRACE to locate the approximate position of the vertex.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 66 of 71
Columbus Public Schools 7/20/05
Polynomial, Power, Rational- D
4. Remember that the vertex form of the equation of a parabola is y = a(x − h)2 + k . What is h
for your parabola?
What is k for your parabola?
5. Quit the application and return to the home screen. Make sure that Plot 1 is On and that L1
and L2 are chosen for a scatterplot. Guess a number for a and enter y = a( x − h )2 + k into
the Y= menu of your calculator, using the vertex for h and k and your guess for a. Check
your guess by graphing your equation with the stat plot. If your parabola does not match
your stat plot, make another guess for a. Keep guessing until the graphs are nearly identical.
Give your equation here.
How did you decide how to change your guess for a to make your graph match?
6. The data from your parabola are stored in L1 and L2. Your calculator can find an equation
that models your data. Such an equation is called a regression. To calculate a quadratic
regression, push STAT and arrow to the right to highlight CALC. Choose QuadReg. DO
NOT PUSH ENTER. With QuadReg on the calculator, on the same line, enter L1, L2. Press
enter. (L1 is 2nd 1 and L2 is 2nd 2. The comma is the key above the 7.) Write the equation
here.
Enter the equation into Y2 and graph. How well does it match your data?
1 2
gt + v0t + s0 ,
2
where g is the acceleration of gravity, v0 is the initial velocity, and s0 is the initial height.
Can you make any connections to the equation of your parabola?
7. Look up the acceleration of gravity. The formula for a falling object is y =
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 67 of 71
Columbus Public Schools 7/20/05
Keep on Bouncing
Answer Key
Polynomial, Power, Rational- D
Materials Needed:
CBR (Ranger)
TI-82,\TI-83, or TI-84 Calculator
Link Cable
Ball
INSTRUCTIONS:
A. Setting up the calculator and Ranger
1. If you are using a TI-83 Plus, skip steps 1 and 2 and go to step 3.
If you are using a TI-82 or TI-83: Select PRGM on the keypad. If the program RANGER is
on the list select it and go to step 4.
2. If RANGER is not on the list, connect your calculator to the Ranger. On the calculator,
select 2nd Link. (It’s on the X key). Use the right arrow to highlight RECEIVE and hit
ENTER. The calculator will display Waiting… Open the RANGER and push the key 82/83.
The calculator should display DONE. Go to step 4.
3. On the TI-83 Plus or TI-84, choose APPS and choose CBL/CBR. (If it is not on the list,
follow the instructions for the 82 or 83.) Press any key. On the next screen select RANGER.
Go the step 4.
4. Hit ENTER. Select #3 Applications. When prompted for UNITS, select #2 FEET. Choose
#3, Ball Bounce.
C. Ball Bounce.
1. Be sure that the ball is dropped on a smooth, level surface. Do not allow anything to obstruct
the path between the Ranger and the ball while the data is being collected.
2. Follow the instructions on the calculator.
3. Your data should look like a series of parabolas, decreasing in height. Decide if you want to
try again or not.
4. Hit ENTER. If you did not like your graph, select #5, REPEAT SAMPLE and go back to
step #2. If you like your graph, go to step 5.
5. Choose #4, PLOT TOOLS. On PLOT TOOLS choose #1, SELECT DOMAIN. Pick out
your best parabola. For LEFT BOUND? use the right or left arrow to move the cursor to the
lowest point on the left side of the parabola you chose. Hit ENTER. For RIGHT BOUND,
use the right arrow to move to the lowest point on the right side of your parabola. Hit
ENTER.
Analysis
1. On your graph, what is measured on the x-axis? time elapsed after the ball is dropped
What is measured on the y-axis? height of the ball at each time
2. The ball was bouncing straight up and down. Why is the graph a series of parabolas? What
force makes the ball fall after each bounce? Why do the heights of the bounces decrease for
each bounce? Because the height function for a falling body is a quadratic. Gravity.
The ball loses energy on each rebound.
3. Use TRACE to locate the approximate position of the vertex. Answers will vary
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 68 of 71
Columbus Public Schools 7/20/05
Polynomial, Power, Rational- D
4. Remember that the vertex form of the equation of a parabola is y = a(x − h)2 + k . What is h
for your parabola?
What is k for your parabola?
5. Quit the application and return to the home screen. Make sure that Plot 1 is On and that L1
and L2 are chosen for a scatterplot. Guess a number for a and enter y = a( x − h )2 + k into
the Y= menu of your calculator, using the vertex for h and k and your guess for a. Check
your guess by graphing your equation with the stat plot. If your parabola does not match
your stat plot, make another guess for a. Keep guessing until the graphs are nearly identical.
Give your equation here.
.
Answers will vary
How did you decide how to change your guess for a to make your graph match?
Answers will vary
6. The data from your parabola are stored in L1 and L2. Your calculator can find an equation
that models your data. Such an equation is called a regression. To calculate a quadratic
regression, push STAT and arrow to the right to highlight CALC. Choose QuadReg. DO
NOT PUSH ENTER. With QuadReg on the calculator, on the same line, enter L1, L2. Press
enter. (L1 is 2nd 1 and L2 is 2nd 2. The comma is the key above the 7.) Write the equation
here.
Answers will vary
Enter the equation into Y2 and graph. How well does it match your data?
1 2
gt + v0t + s0 ,
2
where g is the acceleration of gravity, v0 is the initial velocity, and s0 is the initial height.
Can you make any connections to the equation of your parabola? Answers will vary
7. Look up the acceleration of gravity. The formula for a falling object is y =
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 69 of 71
Columbus Public Schools 7/20/05
Adventures in Multiplicity
Polynomial, Power, Rational- Rtch
Name
For each of the following groups of factored polynomial functions, use the TI-83 to graph them.
Then describe what is different about the polynomials and what is different about the graphs.
Group 1
A) y = x ( x + 3)( x − 2)
B) y = x 2 ( x + 3)( x − 2)
C) y = x( x + 3) 2 ( x − 2)
Observations:
Group 2
A) y = ( x + 4)( x − 1)( x − 3)
B) y = ( x + 4) 2 ( x − 1)( x − 3)
C) y = ( x + 4)3 ( x − 1)( x − 3)
Observations:
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 70 of 71
Columbus Public Schools 7/20/05
Adventures in Multiplicity
Answer Key
Polynomial, Power, Rational- Rtch
For each of the following groups of factored polynomial functions, use the TI-83 to graph them.
Then describe what is different about the polynomials and what is different about the graphs.
Group 1
A) y = x ( x + 3)( x − 2)
B) y = x 2 ( x + 3)( x − 2)
C) y = x( x + 3) 2 ( x − 2)
Observations:
The power on the factors are different.
Whether the graph touches the x-axis or crosses the x-axis is different.
Group 2
A) y = ( x + 4)( x − 1)( x − 3)
B) y = ( x + 4) 2 ( x − 1)( x − 3)
C) y = ( x + 4)3 ( x − 1)( x − 3)
Observations:
If the power of the factor is odd, then the graph crosses the x-axis at that zero. If the power
of the factor is even, then the graph touches the x-axis at that zero.
PreCalculus Standards 4 and 5
Polynomial, Power, & Rational Fctns
Page 71 of 71
Columbus Public Schools 7/20/05