Unit 1: Honors Precalculus
 Lesson 1: Standard 1.1 and 1.2 (1-1, 1-2)
 Lesson 2: Standard 1.3 (1-3, 1-4)
 Lesson 3: Standard 1.3 and 1.4 (1-5, 1-7)
 Lesson 4: Standard 1.3 (1-6)
 Lesson 5: Standard 1.6 (1-8, 2-6)
 Lesson 6: Standard 1.5 (2-1, 2-2)
Welcome to Precalculus!
Mrs. Bunting
Room C109
Get an index card and your handouts.
Pick up a textbook (Rust with spiral on front)
Find your seat on the seating chart and take your
seat. Fill out your index card.
Please begin to work on reviewing the material in
Section 1-1 of your book.
Use your textbook and tablemates to help yourself
review this material.
You will need to TAKE NOTES on the material.
Complete p 10 #17 – 37 odd, 41-47 all
Standard 1.1: distinguish between relations and
functions, identify domain and range, and evaluate
functions (Section 1-1)
p 10 #17 – 37 odd, 41-47 all
By the time you and your group finish you will answer…
 What is a relation?
 What is contained in the domain of a relation?




In the range?
What is a function and how is it different from a
relation?
What is the vertical line test and what is it used for?
What does function notation look like?
How are functions evaluated for specific values?
What is Honors Precalculus?
 You will be introduced to:
Higher level algebra skills!
 Common and Natural Logarithms!
 Limits!
 Arithmetic, Geometric and Infinite Series!
 Polynomial, Rational and Exponential Functions!
 Lots of Trigonometry!
 Rectangular and Polar Coordinates!

not necessarily in that order…
What Can I Expect?
 We will cover at least a section a day.
 We will complete a unit pretty much weekly.
 Each quarter will have several portfolio projects.
 You can expect to have Precalculus work to do
every single night.
Grading to Standards:
 In a nutshell:


1. You need to master EVERY standard to pass.
2. Any standard which you do not pass must be
reassessed.
 To Do Well:


1. Complete your homework.
It is your ticket to reassess.
2. Reassess promptly while things are fresh.
Extra Help
 Tutorial – right here in C109!
Everyday but Tuesday (Library Duty)
 Got Math?
 3C
in C211 Ms Kielkucki
 3D in C106 Ms Ciliano
 4C in C104 Mr. Lisella
 4C in C100 Ms Rohrer
 4D in C100 Ms Bunting
Unit 1: “Get in Line”






In this unit we will complete…
Standard 1.1: distinguish between relations and functions,
identify domain and range, and evaluate functions (1-1)
Standard 1.2: perform operations (add, subtract, multiply,
divide, compose) on functions (1-2)
Standard 1.3: analyze graphs and make predictions based on
linear functions (1-3, 1-4, 1-5, 1-6)
Standard 1.4: graph and interpret piecewise functions (1-7)
Standard 1.5: solve systems of equations (2-1, 2-2)
Standard 1.6: solve systems of linear inequalities (1-8, 2-6)
Standard 1.1 (continued):
Using the Vertical Line Test and Stating
The Domain From A Graph (1-1)
y
y
x
x
y
y
x
x
Standard 1.1: Finding the Domain of
a Function in Equation Form (1-1)
 To find out what the independent (x) values for a
function will be involves finding out what they
cannot be.
 There are TWO Bozo No-No’s:
 No values which cause zero’s in denominators
 No values which cause a negative under a square root
(or any even root)
Find the values for x which are not
in the domain of the function, then
state the domain in proper set
notation.
f (x ) 
x
2
x 1
Find the values for x which are not
in the domain of the function, then
state the domain in proper set
notation.
f (x )  x  3
Find the values for x which are not in the domain
of the function, then state the domain in proper set
notation.
f (x )  x  3x  2
2
Find the values for x which are not
in the domain of the function, then
state the domain in proper set
notation.
f (x ) 
3x
x 5
2
Standard 1.2: perform operations
(add, subtract, multiply, divide, compose)
on functions (1-2)
When we finish this lesson you will be able to …
• Perform basic math operations with
functions
• Create, use and check composite functions
Given:
f (x )  2x  2 and g (x )  x  1
2
 Add the functions:

 Written:
(f  g )(x )
 It means:
(f  g )(x )  f (x )  g (x )
Given:
f (x )  2x  2 and g (x )  x  1
2
 Subtract the functions:

 Written:
 It means:
(f  g )(x )
(f  g )(x )  f (x )  g (x )
Given:
2
f (x )  2x  2 and g (x )  x  1
 Multiply the functions:
 Written:
(f  g )(x )
 It means:
(f  g )(x )  f (x )  g (x )
Given:
f (x )  2x  2 and g (x )  x  1
2
 Multiply the functions:

f
Written: 
g
 It means:
f

g

 (x )


f (x )
, g(x)  0
 (x ) 
g (x )

You try it…
 Given:
2x
f (x )  x  3 and g (x ) 
x 5
Find: (f  g )(x )
Composite Functions:
f
g (x )  f  g (x ) 
 Careful with notation, this is not multiplication.
 It means you actually put one function into the other.
 The second one is going into the first.
Example:
1
If f (x )  x  1 and g (x ) 
x 1
Find f
g  (x )
Example:
1
If f (x )  x  1 and g (x ) 
x 1
Find g f  ( x )
To Check:
f
g (3)  f  g (3) 
Homework:
 For Tomorrow:
 HW 1.1: p 10 #17 – 47 odd, 48-50 all
 HW 1.2: p 17 #11 – 23 odd, 31
 By Monday:
 Cover book
 Get your binder or notebook setup
 Get parental form turned in
Warm-Up:
 P 25 #41
 Have your homework out to be checked!
Homework:
Standard 1.3:
analyze graphs and make
predictions based on linear functions (1-3,1-4)
At the end of this lesson you will be able to…
 Identify and properly use the three forms of
linear equations
 Find x- and y-intercepts
 Define, identify and use the formula for slope
 Identify the two special cases of slope
Linear Functions
 What does a linear equation look like?
 Are all the equations of lines also functions?
 How many of the forms do you remember?
Standard Form:
Ax  By  C or Ax  By  C  0
Standard Form:
Ax  By  C or Ax  By  C  0
 Where A, B and C are numbers like this.
2x  6y  18
 In this form you can tell what about the line?
 Nothing.
Slope-Intercept Form
y  mx  b
 Where m is…
 And b is…
 In this form you can…
 Tell exactly what the line looks like
 Graph the line
Point-Slope Form:
y  y1  m  x  x1 
 Used to develop the linear equation if you know the
slope, m, and one point on the graph, (x1, y1).
 Find the standard form of the equation of the line
which has a slope of -1 and passes through the point
(-4, 5).
What if you only have two points on the graph?
y2  y1
m
x2  x1
 Find the standard form
of the equation which
passes through the
points (6,5) and (4,-5).
 Find slope.
 Use slope and one of the
points to find equation
of the line.
Graph a couple…
2
y  x 4
3
6x  4y  12
The Two Special Cases of Slope:
Finding the Zero of a Linear Function:
7x  3y  21
 Zero is another name for the x-intercept. You
will also hear it called a root.
 The y-intercept is called b but not much else.
Finding x- and y-intercepts:
Homework:
 HW1 1.3: P24 #13 – 33 every other odd
 HW2 1.3: P30 #11 – 27 every other odd
Warm-up:
Homework:
Standard 1.3: analyze graphs and make
predictions based on linear functions (1-5)
By the end of this lesson we will be able to answer…
 How can parallel and perpendicular lines be
identified from their equations?
 How can the properties of lines be used to identify
geometric figures?
 How can the coefficient for an equation be found so
that it will be parallel or perpendicular to a specific
line?
Parallel and Perpendicular Lines
 Parallel lines have the same slope
 Perpendicular lines have slopes which are
negative reciprocals of each other.
1
y  x 3
2
 Find the equation of the line parallel to the
equation above and passing through (2,-2)
 Find the equation of the line perpendicular to the
equation above and passing through (-4,1)
Special Case:
x  3y  5
7x  21y  35
 Lines which have the same slope and the
same y-intercept are called coinciding.
Slope and Distance:
 Consider the polygon with vertices at (0,0), (1,3),
(3,-1) and (4,2).
 Is it a parallelogram?
 Is it a rectangle?
Are these lines parallel, coinciding
perpendicular or none of these?
1. y  5x  5
y  5x  2
neither
2. y  6x  2
3. y  x  6
x y 8  0
parallel
4. y  2x  8
1
y  x 8
6
4 x  2y  16  0
perpendicular
coinciding
Solving for an unknown coefficient:
For what value of k is the graph of kx  7 y  10  0
parallel to the graph of 8x  14y  3  0?
Solving for an unknown coefficient:
For what value of k is the graph of kx  7 y  10  0
parallel to the graph of 8x  14 y  3  0?
For what value of k is it perpendicular?
Standard 1-4: graph and interpret
piecewise functions
(1-7)
In this lesson we will …
 Identify piecewise functions including
greatest integer, step and absolute value.
 Graph piecewise functions.
Piecewise Functions:
 Different equations
are used for different
intervals of the
domain.
 The graphs do not
have to connect.
Graphing Piecewise Functions:

2, if x  5

f (x )  x  4, if  5  x  4
 1
 x , if x  4
 2
Another…
x  3, if x  0

h (x )  3  x , if 0  x  3
3x  3, if x  3

Step Functions:
 Are piecewise
functions whose
graphs look like a set
of steps.
 One example of a step
function is the
greatest integer
function.
f (x )  x
An example of a step function
fee schedule:
 The cost of mailing a
letter is $0.37 for the
first ounce and $0.23
for each additional
ounce or portion
thereof.
The Absolute Value Function:
f (x )  x
Make sure you find the turning point:
f (x )  2 x  3
Make sure you find the turning point:
f (x )  2x  5  2
Homework:
 HW3 1.3: p36 #13-31 odd
 HW 1.4: p49 #11-33 odd
Warm-up: Feel free to get a piece of graph
paper from the bin by the windows.
 Graph the functions:
3 if  1  x  1

h (x )  4 if 1  x  4
x if x  4

x  3, if x  0

r (x )  3  x , if 0  x  3
3x  3, if x  3

Homework:
Standard 1.3: analyze graphs and make
predictions based on linear functions (1-6)
In this section we will…
 Draw and analyze scatter plots.
 Draw a best-fit line and write a prediction
equation.
 Solve problems using prediction equation models.
Collecting and Using Data:
 Real life data seldom forms nice straight lines or
smooth curves.
 For graphs which approximate a line, a best-fit line
(also called a regression line) can be drawn and a
prediction equation can be determined.
Scatter Plots: p 38
 Basically, data is the
graph of a relation.
 If the graph shows a
linear trend you can
create a prediction
equation.
 Accuracy of predictions
depends on how closely
the data approximates a
line.
Correlation:
p 40
 This refers to how closely a set of data actually
approximate a line.
 If the data is very scattered, that is a weak
correlation.
 If the data is very close to being on a line then it has
a strong correlation.
 Our example had moderate correlation.
More About Correlation:p 40
 Correlation is measured using a correlation
coefficient (r).
 r < ½ means weak, ½ < r < ¾ is moderate, ¾ < r <
1 is strong.
 One means complete correlation.
 NOTICE: r is positive for positive slopes and
negative for negative slopes.
The Prediction Equation:
 Graph your data.
 Draw a best-fit line.
 Chose two points, on
the line.
 Find their slope.
 Use the slope and one
of the points to find the
prediction line.
Regression Lines on the Calculator:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Go to STAT, choose EDIT, and enter the x-values in L1 and the y-values in L2.
Go to STAT PLOT (2nd, Y=), press ENTER on 1:Plot 1, and turn Plot1 On.
Go to WINDOW, and adjust your Xmin, Xmax, Ymin, and Ymax to fit your
data.
Go to GRAPH to see your points plotted.
Go to STAT, choose CALC, arrow down to highlight the appropriate regression
model, and press ENTER. Press L1 (2nd, 1), the comma (above the 7), L2 (2nd, 2),
the comma again, then VARS, choose Y-VARS, choose Function, choose Y1,
and press ENTER.
Go to Y= to see that your equation has been transferred to the Y= screen.
Go to GRAPH to see your line.
To enter an x-value and find the corresponding y-value, go to CALC (2nd,
TRACE) and choose 1:value. Enter the x-value, and the y-value will be provided.
To enter a y-value and find the corresponding x-value, go to Y= and next to Y2
graph the line y=a, where a is the y-value in which you are interested. Then go to
CALC (2nd, TRACE) and choose 5:intersect. Press ENTER three times, and
the point of intersection will be provided.

 NOTE: You may need to change your viewing window to accomplish steps 8 and 9.
Now…do it yourselves.
 Use the data your group was given.
 Paste the chart with your data and plot your points




on the large sheet of graph paper.
Draw a best-fit line.
Choose two points on your line and determine your
prediction equation. Show all work on the graph
paper. Label it “Hand Calculated Equation”
Finally, use the graphing calculators to find the
regression equation. Record it on the graph paper
and label it “Calculator Generated Equation”.
Make sure that you allow enough room on the paper
to answer your questions.
Homework:
HW5 1.3: p42 #7 and 9
Warm-up:.
 Grab a couple pieces of graph paper for the lesson.
Feel free to hole punch it
 For heaven’s sake! Finish those projects!
Homework:
Standard 1.6:
Graph and solve using linear inequalities (1-8)
In this section we will…
• Graph linear inequalities
• Graph more complex inequalities
Inequality Graphs:
 Any line will cut the
coordinate plane into
two halves.
 Any point on the line
will cause the
statement to be true.
y  x 3
Inequality Graphs:
 Any point above the line
causes...
y
x 3
 Any point below the line
causes...
y
x 3
Let’s try this one…
2 x  3 y  12
How about this?
7 x y 9
Or this?
y  x 1
Okay Partners, wrangle these…
 You will need graph paper.
 p 55
 Partner 1 graphs #12, Partner 2 graphs #10
 Switch papers and check each other.
 Partner 1 graphs #14, Partner 2 graphs #18
 Switch papers and check each other.
Answers:
Standard 1.6:
Solve systems of linear inequalities (2-6)
At the end of this section you should be able to …
 Find the solution for a system of inequalities using
a graph
 Graph a polygonal convex set
 Find the vertices for a polygonal convex set
 Find the minimum and maximum values for a
polygonal convex set
How can the solution for a system of
inequalities be determined using a graph?
f (x , y )  4x  3y
4y  x  8
x y 2
y  2x  5
What is a polygonal convex set?
 A polygonal convex set is the solution for a system of
inequalities.
 The solution is contained within the polygon formed
by the boundaries of the inequalities.
First graph the inequalities and determine the
polygonal convex set.
f (x , y )  4x  3y
4y  x  8
x y 2
y  2x  5
How do I find the vertices for a polygonal
convex set?
f (x , y )  4x  3y
4y  x  8
x y 2
y  2x  5
How can I find the minimum and maximum
values for a polygonal convex set?
f (x , y )  4x  3y
4y  x  8
x y 2
y  2x  5
Try this one…
f (x , y )  3x  4 y
x  2 y  7
x y 8
2x  y  7
(3,-11)
Word Problem!!! P111 #26
One more!
 The Cruiser Bicycle Company makes two styles of
bicycles: the Xenon, which sells for $200, and the
Yaris, which sells for $600. Each bicycle has the
same frame and tires, but the assembly and
painting time required for the Xenon is only 1
hour, while it is 3 hours for the Yaris. There are
300 frames and 360 hours of labor available for
production. How many bicycles of each model
should be produced to maximize revenue, and how
much money will be made?
Homework: Grab graph paper!
 HW1 1.6: P55 #9 – 21 every other odd and #23
 HW2 1.6: P110 #9 – 21
 HW3 1.6: P117 #15
 Look for a Unit 1 Test on Tuesday 2/15!!!
 Portfolio 1 due on Wednesday 2/16!!!
Warm-up:
Homework:
Standard 1.5:
solve systems of equations (2-1, 2-2)
In these sections we will…
 Solve systems of equations involving two variables
algebraically.
 Solve systems of equations involving three variables
algebraically.
 You will need a ruler and a piece of graph paper.
What does the solution for a system of
linear equations represent?
 3,5 is a solution for the system below:
y  x  8
y  4x  7
How Can We Solve a System?
 Graphing
 Elimination
 Substitution
Solve the following systems by graphing:
1. x  y  2
3x  y  10
2. x  3y  0
2x  6y  5
Terminology:
 If lines intersect: ONE solution
a.k.a. consistent and independent
 If same line twice: INFINITE solutions
a.k.a. consistent and dependent
 If lines are parallel: NO solution
a.k.a. inconsistent
 What were your graphs?
Substitution and Elimination:
1. 5x  y  16
2x  3y  3
2.  35x  40y  55
7 x  8y  11
Word Problem!
 p 71 #10
Solving Systems in 3 Variables
 A system in 3 variables represents the intersection
of 3 planes.
 Look at page 73.
 You need 3 equations to solve.
 You have to have the same number of equations as
you have variables.
 Solve using substitution or elimination.
Let’s try some…
3 y  9 z
4x  2 y  2z  0
3 x  y  4 z  2
How should the solution be written?
x  3y  7z  14
5x  3y  2z  3
2x  4y  z  7
5x  2y  z  11
2x  y  3z  0
6x  2y  2z  16
4x  3y  2z  12
x y z 3
2x  2y  2z  5
Now…YOU think.
 Write a system of 3 equations that fits each
description.

The system has a solution of x = - 5, y = 9 and z = 11.

There is no solution to the system.

The system has an infinite number of solutions.
Homework:
 HW1 1.5: P 71 #22 – 25 all
 HW2 1.5: P 76 #9, 11 and 13
 UNIT 1 Test on Tuesday 2/15
 Portfolio 1 due Wednesday 2/16