Precalculus
Chapter 1
Linear Functions and
Their Graphs
Prange, T. Iverson, Zurek
1
Blank Page
2
Chapter.
Section.
Target#
Learning Target
1.1.1
I can find the slope given two points.
1.1.2
I can find the slope given an equation of a
line.
Practice for the
Learning Target
Score on
Learning
Target Quiz
Help
needed?
yes/no
Pg 11 7 (slope
only), 9 (slope only)
Pg 12 19a, 20a
Pg 11 5 (identify the
y-intercept), 6
(identify the yintercept), 23a
1.1.3
I can determine the y-intercept from a
graph or an equation of the line.
1.1.4
I can interpret what the slope, y-intercept,
and x-intercept mean in a real life
situation.
1.1 worksheets in
this packet
1.1.5
I can write the equation of a line in pointslope form and slope-intercept form.
Pg 12
25 (slope intercept
form only), 33
(point slope form
only)
1.1.6
I can write the equation of a parallel line.
Pg 12 55a, 57a
1.1.7
I can write the equation of a perpendicular
line.
Pg 12 55b, 57b
1.1.8
I can write, and graph, the equation of
vertical and horizontal lines.
Pg 11-12 1b, 2a,
4d, 29, 31
1.1.9
I can write the equation of a line that
models a real world situation.
1.1 worksheets in
this packet
1.7.1
I can find the regression line equation
given a set of points.
Pg 77 7(part b-d),
12, 13, 14, 16
1.7.2
I can analyze, interpret, and predict using
the regression line.
Pg 77 12, 13, 14, 16
1.5.1
Given an equation written in function
notation, I can evaluate the function
algebraically.
1.5 worksheet in this
packet
1.5.2
I can evaluate a function using the graph.
1.5 worksheet in this
packet
3
Chapter.
Section.
Target#
Learning Target
1.3.1
I can graph a linear piecewise function
from an equation or scenario.
1.3.2
Given a piecewise function, I can find the
value of y, algebraically.
1.3.3
I can write a piecewise function from a
graph.
Practice for the
Learning Target
Score on
Learning
Target Quiz
Help
needed?
yes/no
Pg 39 41, 45
Pg 83 65
Evaluate all three
problems for f(1).
Pg 39 41, 45
Pg 83 65
Evaluate all three
problems for f(1).
1.3 worksheet in this
packet
Pg 38 11-16
Determine if the
graph represents a
function using the
Vertical Line Test.
1.3.4
I can identify whether or not a graph
represents a function.
1.3.5
I can identify the domain and range from a
piecewise graph using interval notation.
1.3 worksheet in this
packet
1.3.6
I can identify and describe increasing,
decreasing, and constant intervals using
interval notation.
1.3 worksheet in this
packet
Essential Questions for the chapter
1. How can patterns, relations, and functions be used as tools to best describe and help explain real-life
situations?
2. How can the collection, organization, interpretation, and display of data be used to answer questions?
Essential Questions for the course
1. How is this similar or different from what I have done before?
2. What can I do to retain what I have learned?
3. Does my answer make sense? If not, what do I do?
4. Do I need help, and where do I go to find it?
5. How would a calculator make this problem easier to do?
6. How do I explain or justify my work to myself and others?
7. What is the given information and how do I use it?
4
SCORING RUBRIC
A+
MASTERY (+)
100% I completely understand the strategy and mathematical operations to be used, and I used them
or
correctly.
 I did all of my calculations correctly.
4.0




My work shows what I did and what I was thinking while I worked the problem.
The way I worked the problem makes sense and is easy for someone else to follow.
I followed through with my strategy from beginning to end.
My work was clear and organized.
M92%
or
3.7
MASTERY (-)
I completely understand the strategy and mathematical operations to be used, but one minor
error kept me completing the problem correctly.
DM
85%
or
3.4
DEVELOPING MASTERY
I understand the strategy and mathematical operations to be used, but a few minor errors kept
me from completing the problem correctly.



My thought process was correct but one minor error kept me from getting the correct answer,
BUT:
o The way I worked the problem makes sense and is easy for someone else to follow.
o I followed through with my strategy from beginning to end.
o My work was clear and organized.
I correctly understood the concept, but my work lacks a few minor elements that would have
made my thought process easy for anyone to follow.
My thought process was correct but a few minor errors kept me from getting the correct
answer.
BU
75%
or
3
BASIC UNDERSTANDING
I used mathematical operations and a strategy that I think works for most of the problem.
IU
50%
or
2
INCOMPLETE UNDERSTANDING
I wasn’t sure which mathematical operations to use, and my plan didn’t work.
NE
0%
or
0
NO EVIDENCE
I did not demonstrate any understanding of the concept.











My work included an obvious conceptual mistake.
Several elements need to be added for my work to be easy to follow.
I know which operations I should have used, but couldn’t complete the problem.
I’m not sure how much detail I need in order to help someone understand what I did.
I made several significant calculation errors.
I tried several things related to the learning target(s), but didn’t get anywhere.
I was not able to reach an answer.
I left the problem blank.
I didn’t know how to begin.
I don’t know what to write.
I wrote down information not related to the learning target(s).
5
Blank Page
6
1.1 Warm Up(s)
7
Date _______
Notes: 1-1 Day 1 & 2
Essential Questions:
1. How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations?
Learning Targets:
1.1.1
I can find the slope given two points.
1.1.2
I can find the slope given an equation of a line.
1.1.3
I can determine the y-intercept from a graph or from the equation of the line.
1.1.4
I can interpret what the slope, y-intercept and x-intercept mean in a real life situation.
1.1.5
I can write the equation of a line in point-slope form and slope-intercept form.
1.1.6
I can write the equation of a parallel line.
1.1.7
I can write the equation of a perpendicular line.
1.1.8
I can write, and graph, the equation of vertical and horizontal lines.
Vocabulary
Slope
parallel line
y-intercept
perpendicular line
x-intercept
vertical line
point slope form
horizontal line
What Do You Already Know?
Choose the best answer that corresponds to each:
_____ 1. Slope-Intercept Form
_____ 2. Vertical Line
_____ 3. Point-Slope Form
_____ 4. Horizontal Line
_____ 5. A line going through (3, -4) and where m is undefined
_____ 6. The slope of a horizontal line
Other Things To Know!
7. Slope formula for two points:
5
9. Simplify 0
slope intercept form
A. x  #
B. y  #
C. y = -4
D. x = 3
E. ( y  y1 )  m( x  x1 )
F. y  mx  b
G. zero
H. undefined
8. Find the slope of the line that contains (-1,5) and (3,-4).
0
10. Simplify 5
11. What is the slope of a vertical line?
Example 1) Find the slope, y-intercept, and sketch the graph of:
-4x - 2y – 8 = 0
Steps:
You try:
a)  x  3 y  12  0
b) 16  8 y  0
8
Example 2) Write the equation of the line containing (-2, 3) & (3, -7) in both forms.
Point-Slope Form:
Steps
Slope-Intercept Form:
You try: Write the equation of the line containing (-3, 1) & (5, 5) in both forms.
Point-Slope Form:
Slope-Intercept Form:
Parallel Lines:
Example 3) Find the point-slope equation of the line that passes through the point (2, -1) and is ∥ to
2𝑥 − 3𝑦 = 5.
Steps:
You Try: Find the slope-intercept form of the line that passes through the point (-3, 1) and is ∥ to
𝑥 + 𝑦 = 5.
9
Perpendicular Lines:
Example 4) Find the point-slope equation of the line that passes through the point (2, -1) and is  to
2𝑥 − 3𝑦 = 5.
Steps:
You Try: Find the slope-int. form of the line that passes through the point (-3, 1) and is  to
𝑦 − 3 = −2𝑥.
10
Homework for 1.1
Day 1: pg. 11-12: 1b, 2a, 4d, 5 (identify the y-intercept), 6 (identify the y-intercept), 7 (slope
only), 9 (slope only), 19a, 20a, 23a, 29, 31
Day 2: pg. 11-12: 25 (slope intercept form only), 33 (point slope form only), 55a, 57a, 55b, 57b
(B.O.B. is wrong for 57b)
11
Continue the 1.1 Homework here
12
Date _______
Notes: 1-1 Application problems Day 1
Essential Questions:
1. How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations?
Learning Targets:
1.1.9
I can write the equation of a line that models a real world situation.
Example 1
A) Make a table of values for the linear function y = 2x + 1.
B) Does the x-value depend on the y-value we input, or does the y-value depend on the x-value
that we input?
C) Write the ordered pairs from the table of values in the space below.
D) Graph the ordered pairs and draw the line that represents the function.
E) If this was the linear function that represents the growth of a dolphin, what other information
would be important to know about our data?
F) So if we were then told that the variable x = ___________________________ and the variable
y = ________________________, what would the slope mean in the context of this problem?
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G) What would the y – intercept mean?
H) Would negative x – values make sense in this situation?
I) What would be the growth per year if our data was:
X
1
4
Y
5
10
Example 2 Find the indicated information for the following application problems:
A) A climber is on a hike. After 2 hours he is at an altitude of 400 feet. After 6 hours, he is at an
altitude of 700 feet. Determine the dependent and independent variables and write the ordered
pairs.
B) What is the average rate of change in altitude?
C) A scuba diver is 30 feet below the surface of the water 10 seconds after she entered the water
and 100 feet below the surface after 40 seconds. Write the ordered pairs and determine the scuba
divers rate of change in depth.
D) A person weighed 145 pounds in 1986 and 190 pounds in 2007. What was the rate of change
in weight?
14
Homework for 1.1 Day 1 Application Problems
1. Over the last 50 years, the average temperature has increased by 2.5 degrees worldwide (I made this
up). What is the rate of change in worldwide temperatures per year?
2. Michael started a savings account with $300. After 4 weeks, he had $350, and after 9 weeks he had
$400. What is the rate of change of money in his savings account per week?
3. A plane left Chicago at 8:00 A.M. At 1:00 P.M., the plane landed in Los Angeles, which is 1500
miles away. What was the average speed of the plane, in miles per hour, for the trip?
4. After 30 baseball games, A-Rod had 25 hits. If after 100 games he had 80 hits, what is his average
hits per game?
For problems 5 – 8 it may be helpful to draw a picture:
5. When the dependent variable increases and the independent variable increases, the rate of change is
(Positive, Negative, zero, undefined) circle one.
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Continue the Homework for 1.1 Day 1 Application Problems
6. When the dependent variable stays the same as the independent variable increases, the rate of change
is (Positive, Negative, zero, undefined) circle one.
7. When the dependent variable decreases when the independent variable increases, the rate of change is
(Positive, Negative, zero, undefined) circle one.
8. When the dependent variable increases when the independent variable stays the same, the rate of
change is (Positive, Negative, zero, undefined) circle one.
16
Date _______
Notes: 1-1 Application problems Day 2
Essential Questions:
1. How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations?
Learning Targets:
1.1.9 I can write the equation of a line that models a real world situation.
Use x = the number of years since 2000.
Example 1 A) During 2002, Nike’s net sales were $9.0 billion, and in 2007 net sales were $11.5 billion.
Write a linear equation giving the net sales y in terms of the year x. (Hint: Write two ordered pairs first.)
B) Predict the net sales for the year 2022.
C) Graph on your calculator. Verify your prediction in B was correct.
D) If Sketchers net sales were 3.0 billion in 2002 and 10.5 billion in 2007, determine the
equation of the linear function representing Sketchers’ net sales during this time.
E) Using your calculator, graph this function on the same grid as the Nike function. In the
context of this problem, explain what the y-intercept means for each function.
F) If you were thinking about investing in one of these companies, which company would be
more successful after four years assuming these trends continue? How about 15 years out?
G) In what year were the net sales for the companies approximately equal? Explain in words,
how you would determine the exact time this occurred.
17
Example 2 Your cell phone plan is $59.99 per month with $0.20 for each additional minute over 500
minutes. (Hint: How much would you pay if you used 500 minutes?)
A) What would the y-intercept be if we are only interested in graphing the minutes exceeding
500.
B) Write a linear equation that will compute your monthly cost if you exceed the 500 minutes.
C) How many minutes did you use for the whole month if your bill is $70.39
(note: no taxes/random fees)
Example 3 A school district purchases a high-volume printer, copier, and scanner for $25,000. After
10 years, the equipment is expected to be worth $2000.
A) Write a linear equation to represent the model.
B) Find the value of the equipment after year 5.
C) What is the x-intercept of this function and explain in words what that point represents.
18
Homework for 1.1 Day 2 Application Problems
Use x = the number of years since 2000.
1. A librarian’s salary was $25,000 in 2000 and $27,500 in 2002. Determine two ordered pairs
for this scenario.
A. Assuming the librarian’s salary follows a linear growth pattern, find the slope of the line.
B. Write the equation in slope intercept form.
C. What will the librarian’s salary be in 2006?
2. The earnings per share of Harley-Davidson stock for in 2005 was $2.50 and in 2011 was
$5.50. Find the equation in slope-intercept form that represents this situation.
A. Interpret the meaning of the slope of the equation.
B. What does the y-intercept indicate?
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Continue the Homework for 1.1 Day 2 Application Problems
3. Pinemoor Pizza purchases a used pizza oven for $875 knowing that it will have to be
replaced after 5 years (it will be worthless at this point).
A. Write a linear equation for the value of the oven over the 5 year period.
B. Angelo’s Pizza, across town, offers to buy the oven for $400. At this point Pinemoor has
had the oven for three years. Prove algebraically why Pinemoor should sell the oven to
Angelo’s.
4. Suppose the weight of an airplane (in pounds) is a linear function of the amount of fuel (in
gallons) in its tank. When carrying 18 gallons of fuel, the airplane weighs 2217 pounds.
When carrying 48 gallons of fuel, it weighs 2412 pounds. How much does the airplane
weigh if it is carrying 60 gallons of fuel?
20
Date _______
Notes: 1-1 Application problems Day 3
Essential Questions:
1. How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations?
Learning Targets:
1.1.9
I can write the equation of a line that models a real world situation.
Model Problems
A linear model is a linear equation that represents a real-world scenario. You can write the equation
for a linear model in the same way you would write the slope-intercept equation of a line. The yintercept of a linear model is the quantity that does not depend on x. The slope is the quantity that
changes at a constant rate as x changes. The change must be at a constant rate in order for the
equation to be a linear model.
Example 1 A machine salesperson earns a base salary of $40,000 plus a commission of $300 for every
machine he sells. Write an equation that shows the total amount of income the salesperson earns, if
he sells x machines in a year.
The y-intercept is $40,000; the salesperson earns a $40,000 salary in a year and that amount does not
depend on x.
The slope is $300 because the salesperson’s income increases by $300 for each machine he sells.
Answer: The linear model representing the salesperson’s total income is y = $300x + $40,000.
Linear models can be used to solve problems.
Example 2 The linear model that shows the total income for the salesperson in example 1 is
y =300x + 40,000. (a) What would be the salesperson’s income if he sold 150 machines? (b) How
many machines would the salesperson need to sell to earn a $100,000 income?
(a) If the salesperson were to sell 150 machines, let x = 150 in the linear model; 300(150) + 40,000 =
85,000.
Answer: His income would be $85,000.
(b) To find the number of machines he needs to sell to earn a $100,000 income, let y = 100,000 and
solve for x:
y = 300x + 40,000 Write the linear model.
100,000 = 300x + 40,000 Substitute y = 100,000.
60,000 = 300x Subtract.
x = 200 Divide.
Answer: To earn a $100,000 income the salesperson would need to sell 200 machines.
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You can also use the standard form to write a linear model. Use this form if you are analyzing two
quantities that increase at different rates.
Example 3 At a school play, children’s tickets cost $3 each and adult tickets cost $7 each. The total
amount of money earned from ticket sales equals $210. Write a linear model that relates the number of
children’s tickets sold to the number of adult tickets sold.
Let x = the number of children’s tickets sold and y = the number of adult tickets sold
The amount of money earned from children’s tickets is 3x.
The amount of money earned from adult tickets is 7y.
The total amount of money earned from ticket sales is 3x + 7y, which is
equal to $210.
Answer: 3x + 7y = 210.
Example 4 In Example 3, how many children’s tickets were sold if 24 adult tickets were sold?
If 24 adult tickets were sold, y = 24. Substitute y = 24 into the linear model above:
3x + 7y = 210
3x + 7(24) = 210
3x + 168 = 210
3x = 42
x = 14
Write the linear model.
Substitute y = 24.
Simplify.
Subtract.
Divide.
Answer: 14 children’s tickets were sold.
22
Homework for 1.1 Day 3 Application Problems
1. Lin is tracking the progress of her plant’s growth. Today the plant is 5 cm high. The plant grows 1.5
cm per day.
a. Write a linear model that represents the height of the plant after d days.
b. What will the height of the plant be after 20 days?
2. Mr. Thompson is on a diet. He currently weighs 260 pounds. He loses 4 pounds per month.
a. Write a linear model that represents Mr. Thompson’s weight after m months.
b. After how many months will Mr. Thompson reach his goal weight of 220 pounds?
3. Paul opens a savings account with $350. He saves $150 per month. Assume that he does not withdraw
money or make any additional deposits.
a. Write a linear model that represents the total amount of money Paul deposits into his account
after m months.
b. After how many months will Paul have more than $2,000?
4. The population of Bay Village is 35,000 today. Every year the population of Bay Village increases by
750 people.
a. Write a linear model that represents the population of Bay Village x years from today.
b. In approximately many years will the population of Bay Village exceed 50,000 people?
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Homework for 1.1 Day 3 Application Problems
5. Conner has $25,000 in his bank account. Every month he spends $1,500. He does not add money to
the account.
a. Write a linear model that shows how much money will be in the account after x months.
b. How much money will Conner have in his account after 8 months?
6. A cell phone plan costs $30 per month for unlimited calling plus $0.15 per text message.
a. Write a linear model that represents the monthly cost of this cell phone plan if the user sends t
text messages.
b. If you send 200 text messages, how much would you pay according to this cell phone plan?
7. Ben walks at a rate of 3 miles per hour. He runs at a rate of 6 miles per hour. In one week, the
combined distance that he walks and runs is 210 miles.
a. Write a linear model that relates the number of hours that Ben walks to the number of hours
Ben runs.
b. Ben runs for 25 hours. For how many hours does he walk?
8. A salesperson receives a base salary of $35,000 and a commission of 10% of the total sales for the
year.
a. Write a linear model that shows the salesperson’s total income based on total sales of k dollars.
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b. If the salesperson sells $250,000 worth of merchandise, what is her total income for the year,
including her base salary?
Continue the Homework for 1.1 Day 3 Application Problems
9. Amery has x books that weigh 2 pounds each and y books that weigh 3 pounds each. The total weight
of his books is 60 pounds.
a. Write a linear model that relates the number of 2 pound books to the number of 3 pound books
Amery has.
b. If Amery has 10 3-pound books, how many 2-pound books does he have?
10. Max sells lemonade for $2 per cup and candy for $1.50 per bar. He earns $425 selling lemonade and
candy.
a. Write a linear model that relates the number of cups of lemonade he sold to the number of bars
of candy he sold.
b. If Max sold 90 bars of candy, how many cups of lemonade did he sell?
25
Continue the Homework 1.1 Day 3 Application Challenge Problems
11. A bacteria population doubles every minute. Explain why this population growth cannot be
modeled using a linear equation.
____________________________________________________________________________________
____________________________________________________________________________________
__________________________________________________________________________________
12. Kara used the linear model y = 20,000 + 0.3x to predict her total salary from achieving total sales of
x. What is her base salary? What percent commission does she earn?
____________________________________________________________________________________
____________________________________________________________________________________
13. Correct the Error
Question: The model 2x + 5y = 85 can be used to model how much money Tim spent on x sodas and
y sandwiches. If he bought 15 sodas, how many sandwiches did he purchase?
Solution:
2x + 5(15) = 85
2x + 75 = 85
2x = 10 or x = 2
Tim bought 2 sandwiches.
What is the error? Explain how to solve the problem.
____________________________________________________________________________________
26
____________________________________________________________________________________
1.7 Warm Up(s)
27
Date _______
Notes: 1-7 Day 1 & 2
Essential Questions:
1. How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations?
2. How can the collection, organization, interpretation, and display of data be used to answer questions?
Learning Targets:
1.7.1
I can find the regression line equation given a set of points.
1.7.2
I can analyze, interpret, and predict using the regression line.
Vocabulary
regression line
28
29
Example 1 The table below lists the resting hear rate (in beats per minute) and ages of 12 members of
an aerobics class.
Age
Resting
Heart
Rate
A.
15
20
25
30
30
37
40
42
45
48
55
55
70
68
68
72
65
64
58
62
55
57
54
50
Define the independent and the dependent variables.
B. Using a graphing calculator, make a scatter plot.
C. Using a graphing calculator, determine the line of best fit for this data set. Round to the
nearest thousandths place and write the equation.
D. Graph the line in the same window as the scatter plot.
E. Write the dimensions of an appropriate window to use for this scatter plot.
X min
Y min
X max
Y max
X scale
Y scale
F. Interpret the y-intercept of the regression equation in the context of the problem.
G. Use the linear regression equation to predict the resting heart rate of a 27-year old.
H. Use the linear regression equation to predict the resting heart rate of a 45-year old.
30
Homework for 1.7 Day 2
Pg 77 7(part b-d only), 12, 13, 14, 16
31
1.7 Day 3 Linear Regression Activity
Barbie Bungee Jump
# of rubber
bands
Distance dropped
(to the top head)
1
2
3
4
5
1. Complete the table.
2. What are the independent and
independent variables?
3.
Plot the points on the grid.
4. Find the equation of the regression line.
5. Give an interpretation of the slope.
6. Give an interpretation of the y-intercept.
7. Use the regression line to predict how far she would fall with 10 rubber bands.
8. Barbie wants to bungee jump off of a 215 inch platform. She is a real thrill seeker and wants to drop
as close to the ground as safely as possible. How many rubber bands will she need?
32
1.5 Warm Up(s)
33
Date _______
Notes: 1.5
Essential Questions:
1. How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations?
Learning Targets:
1.5.1
Given an equation written in function notation, I can evaluate the function algebraically.
1.5.2
I can evaluate a function using the graph.
Vocabulary
function notation
Example 1: Given f ( x)  x 2  x  3 and 𝑔(𝑥) =
a) find f (4) .
(𝑥+4)(𝑥−8)
(𝑥+2)
b) find g ( 3) .
c) evaluate 𝑓(𝑥 − 1).
Example 2 Evaluate the function using the graph.
a)
b)
𝑓(3) =
c)
𝑝(0) =
𝑡(−1) =
Example 3 Using function notation to help solve a real-world problem.
You are helping to organize a field trip to Yellowstone National Park for your science class. After
some careful planning, you have discovered two equations that can be used to calculate the cost of
the trip. The cost of lodging is indicated by the equation: 𝐿(𝑥) = 1400 + 15𝑥 where x is the
number of people who come along. Likewise, the daily meal cost can be described by the equation:
𝑀(𝑥) = 3(8 + 7𝑥) where x, again, represents the number of people who make the trip. Describe
using complete sentences how you would calculate the total daily cost of the trip for 7 people, then
go ahead and calculate the cost.
34
Homework 1.5 Worksheet
Use the following functions for problems 1 – 4.
f ( x)  5 x 2  4 x
1
1. Evaluate 𝑓 (2).
g( x)  3 x  1
h( x ) 
7  2x
x 3
k ( x) 
2
2. Find 𝑔(−3.5).
3. Determine the value of 𝑘 (3).
4. Explain why ℎ(3) is undefined.
5. Evaluate 𝑓(𝑥 + 2).
6.
7.
𝑓(3) =
8.
1
2x
𝑝(0) =
9.
𝑡(−1) =
𝑇(2) =
35
Quiz Review (Sections 1.1, 1.7 and 1.5)
1. Given the line 3𝑥 − 4𝑦 = 8
1.1.5 I can write the equation of a line in point-slope form and slope-intercept form.
a) Put the equation in slope-intercept form.
1.1.2 I can find the slope given an equation of a line.
b) Identify the slope of this line.
1.1.6 I can write the equation of a parallel line.
c) Find the equation of the line passing through (2, -1) which is parallel to the original line. Write
your answer in point-slope form.
1.1.7 I can write the equation of a perpendicular line.
d) Find the equation of the line passing through (2, -1) which is perpendicular to the original line.
Write your answer in slope-intercept form.
1.1.1 I can find the slope given two points.
1.1.3 I can determine the y-intercept from a graph or an equation of the line.
2. Find the slope and y-intercept of the line passing through (-3, -1) and (-2, -5).
1.1.2 I can find the slope given an equation of a line.
1.1.3 I can determine the y-intercept from a graph or an equation of the line.
3.
Find the slope and y-intercept of the equation 3𝑥 = 4𝑦 − 7.
1.1.8 I can write, and graph, the equation of vertical and horizontal lines.
4. Write the equation of a line whose slope is zero and y-intercept is 37
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Continue Quiz Review (Sections 1.1, 1.7, and 1.5)
1.1.1 I can find the slope given two points.
1.1.3 I can determine the y-intercept from a graph or an equation of the line.
5. Find the slope and y-intercept of the line.
6. Shelia’s Appliance Repair charges $25 to make a house call and an additional $12 an hour for time
spent at the appointment site.
1.1.9 I can write the equation of a line that models a real world situation.
a) Write an equation that represents the total cost you would expect to pay if Shelia cane to fix your
leaky dishwasher?
1.1.4 I can interpret what the slope, y-intercept, and x-intercept mean in a real life situation.
b) Identify the slope from your equation and explain what it means in the context of this problem.
1.1.4 I can interpret what the slope, y-intercept, and x-intercept mean in a real life situation.
c) Identify the y-intercept from your equation and explain what it means in the context of this
problem.
7) Justin, a linebacker for the University of Kansas Jayhawks, believes that eating M&Ms before the
game causes him to get more tackles during the game. During his previous several games, Justin has
taken the following data.
Number of M&Ms Eaten
25
60
45
80
30
Number of Tackles
3
5
4
7
3
1.7.1 I can find the regression line equation given a set of points.
a) Use your calculator to find a linear regression equation that represents this data.
1.7.2 I can analyze, interpret, and predict using the regression line.
b) How many tackles should Justin expect to get if he eats 150 M&Ms. Assume you cannot have a
fraction of a tackle.
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Continue Quiz Review (Sections 1.1, 1.7, and 1.5)
1.5.1 Given an equation written in function notation, I can evaluate the function algebraically.
8) Given ℎ(𝑥) = 𝑥 2 + 2𝑥 and 𝑘(𝑥) = − |𝑥 + 3| − 7
a) find ℎ(6).
b) find 𝑘(3).
c) find ℎ(𝑥 − 1).
d) find 𝑘 (5).
2
1.5.2 I can evaluate a function using the graph.
9) Evaluate each function using the graph.
a)
b)
t(5) =
c)
g(-2) =
f(3) =
h(-1) =
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Warm Up 1.3
Graph the following absolute value functions. Use a graphing utility to help you remember the translation
patterns from AAT.
1) 𝑓(𝑥) = |𝑥|
3) 𝑔(𝑥) = −|𝑥 − 3|
2) 𝑡(𝑥) = |𝑥 + 2|
4) ℎ(𝑥) = |𝑥 + 5| − 2
Summarize the translations above. You will need to use them for the remainder of this packet.
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Date _______
Notes: 1.3 Day 1 & 2
Essential Questions:
1. How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations?
Learning Targets:
1.3.1
I can graph a linear piecewise function from an equation or scenario.
1.3.2
Given a piecewise function, I can find the value of y algebraically.
1.3.3
I can write a piecewise function from a graph.
Vocabulary
piecewise functions
function
function notation
Definition of a Piecewise-Defined Function:
Why?
Example 1 Graph the following piecewise functions and evaluate the function at a given value.
 f ( x)  3  x x  0
a) 
 f ( x)  x  1 x  0
1

 f ( x)  x  2 x  2
3
b) 
 f ( x)  x  2 x  2

40
Example 2 Write a function that represents each scenario and graph it. (label axis and include scale)
a)
The parking lot charges $3 per hour for the initial
3 hours and $10 after this.
b) Jacob was paid an hourly wage of $8 for 30 hours
per week and an additional hourly wage of $12 for
more than 30 hours.
c)
A copy shop charges $0.15 per photocopy for orders
of 20 or fewer photocopies, $0.12 per photocopy for
orders of 50 or fewer but more than 20 photocopies,
and $0.10 per photocopy for orders of more than 50
photocopies. Write a function represents the cost C
in dollars for x number of photocopies?
d) A store charges $15 per t-shirt for orders of 50 or
fewer T-shirts, $13.50 per t-shirt for orders of 75
or fewer but more than 50 t-shirts, and $12.50 per
t-shirt for orders of more than 75 t-shirts. Which
function best represents the printing cost C for x
number of t-shirts?
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Example 3
a) Draw a graph the shows a man climbing down a ladder that is 10 feet high. At time 0 seconds,
his shoes are at 10 feet above the floor, and at time 6 seconds, his shoes are at 3 feet. From time
6 seconds to the 8.5 second mark, he drinks some water on the step 3 feet off the ground.
Afterward drinking the water, he takes 1.5 seconds to descend to the ground and then he walks
into the kitchen. The video ends at the 15 second mark.
b) Make up an elevation-versus-time graphing story for the following graph:
B
A
C
E
D
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x  2 x  1
Example 4 Given f ( x)  
2 x  3 x  1
and
t  3
t

g (t )   2
t  3 t 1

3
a) find f (2) .
c) evaluate g (3) .
b) find f (1) .
d) determine g (2) .
 x  1 x  0
You Try) Given f ( x)  
 x  1 x  0
e) evaluate f (1) .
f) find f (0) .
Example 5 Write a function that models each graph.
a)
b)
43
c)
d)
44
Homework for 1.3 Day 1 & 2
Pg 39 #41, 45 and Pg 83 #65
Evaluate all three problems for f(1).
41.
45.
f(1) =
f(1) =
65.
f(1) =
45
Homework for 1.3 Day 1 & 2
Write a piecewise equation that represents each of these graphs.
1.
2.
3.
4.
5.
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Date _______
Notes: 1.3 Day 3 & 4
Essential Questions:
1. How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations?
Learning Target:
1.3.4
I can identify whether or not a graph represents a function.
1.3.5
I can identify the domain and range of a piecewise graph using interval notation.
1.3.6
I can identify the increasing, decreasing, and constant intervals using interval notation.
Vocabulary
function
function notation
increasing interval
domain
range
vertical line test
horizontal line test
decreasing interval
constant interval
relative maximum
relative minimum
Definition of a Function:
Vertical Line Test:
Example 1 Using the Vertical Line Test to determine if a graph represents a function:
a)
b)
c) Graph y   x  3  4 and determine if the
equation represents y as a function of x.
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Definition of the Domain of a Function:
Tips: Look for values that you cannot have:
 Can’t have zero in the denominator
 Can’t have negative numbers under the radical
Otherwise, usually, the domain is all real numbers, which is written ___________ in interval notation.
Definition of the Range of a Function:
Tips: Look for values that you cannot have:


Can’t have radicals equaling negative numbers
Can’t have absolute values equaling negative numbers
Otherwise, usually, the range is all real numbers, which is written ___________ in interval notation.
Example 2 Sketch the functions on the grid provided then determine the domain and range. Use the
tips above verify your answer.
a) g ( x)  2 x  3
3
b) f ( x)   x  6
2
Domain:_______________
Domain:_______________
Range:_______________
Range:_______________
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c) f ( x)  x  5
d) p( x)  x  2  3
Domain:_______________
Domain:_______________
Range:_______________
Range:_______________
x  3 x  1
e) g ( x)  

x4 x3
4

1
f) h( x)   x  1
 4
 x  2 +3
x  3
0 x3
x3
Domain:_______________
Domain:_______________
Range:_______________
Range:_______________
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In general, what does an increasing, decreasing, and constant interval look like? How does it
relate to slope?
Example 1 Determine the intervals on which the function is increasing, decreasing or constant. Use
interval notation.
a)
Increasing _______________________
Decreasing _______________________
Constant _________________________
b)
Increasing _______________________
Decreasing _______________________
Constant _________________________
c)
Increasing _______________________
Decreasing _______________________
Constant _________________________
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d)
Increasing _______________________
Decreasing _______________________
Constant _________________________
Example 4 Now try to put all of the elements together. Determine the domain, the range, and the
intervals on which the function is increasing, decreasing or constant. Use interval notation.
a)
b)
Domain _________________________
Domain _________________________
Range ___________________________
Range ___________________________
Increasing _______________________
Increasing _______________________
Decreasing _______________________
Decreasing _______________________
Constant _________________________
Constant _________________________
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Homework for 1.3 Day 3 & 4
Pg 38 11-16 Determine if the graph represents a function using the Vertical Line Test.
Use the graph to determine the following information.
1.
2.
Domain _________________________
Domain _________________________
Range ___________________________
Range ___________________________
Increasing _______________________
Increasing _______________________
Decreasing _______________________
Decreasing _______________________
Constant _________________________
Constant _________________________
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Pre-Calculus Chapter 1 Test Review - You may use a graphing calculator on this review.
1.1.2 I can find the slope given an equation of a line.
1.1.3 I can determine the y-intercept from a graph or an equation of a line.
1. Graph the function 3 x  4 y  12 .
1.
1.1.1 I can find the slope given two points.
1.1.5 I can write the equation of a line in point-slop form and slope-intercept form.
2. Find the equation of the line in slope-intercept form
that passes through the points (-4, 2) and (-8, 5).
2. ______________________
1.1.6 I can write the equation of a parallel line.
3.
Find the equation of the line in slope-intercept form that
1
passes through the point (1, -9) and is parallel to y  x  21 .
3
3. ______________________
1.1.7 I can write the equation of a perpendicular line.
4. Find the equation of the line in point-slope form that
passes through the point (-6, 2) and is perpendicular to 3x  9 y  12 .
4. _____________________
1.1.8 I can write, and graph, the equation of vertical and horizontal lines.
5. Write the equation of a vertical line passing through
the point (3, -2).
5. _____________________
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1.1.9 I can write the equation of a line that models a real world situation.
6. A salesperson receives a base salary of $75,000 and a
commission of 8% of the total sales for the year. Write a
linear model that shows the salesperson’s total income
based on total sales of k dollars.
6. _____________________
Use the following table for questions 7-10. The table shows the sales S (in millions of dollars) for
Timberland from 1995 to 2002. Use x = 0 for 1990.
Year, x
1995
1996
1997
1998
1999
2000
2001
2002
Sales, S
665.1
690.0
796.5
862.2
917.2
1091.5
1183.6
1190.9
1.7.1 I can find the regression line equation given a set of points.
7. Use the regression feature on the graphing calculator
7. _____________________
to find a line of best fit (round to the nearest thousandths)
for the data. Let x represent the year with x = 5 corresponding
to 1995.
1.1.4 I can interpret what the slope, y-intercept, and x-intercept mean in a real life situation.
8. What does the slope represent in the context of the problem? Use the correct units and write
the answer in a complete sentence.
1.7.2 I can analyze, interpret, and predict using the regression line.
9. Use the line of best fit or the table feature on your calculator
to find the year in which the sales will exceed $1300 million.
9. _____________________
1.7.2 I can analyze, interpret, and predict using the regression line.
10. Use the line of best fit to predict the average salary for
the year 2006 (round to the nearest tenths place).
10. _____________________
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1.5.1 Given an equation written in function notation, I can evaluate the function algebraically.
3
1
11. Given the function (𝑥) = − 5 𝑥 + 4 , find 𝑓 (5).
11. _____________________
1.5.2 I can evaluate a function using the graph.
12. Find g(4) using this graph.
12. _____________________
1.3.2 Given a piecewise function, I can find the value of y algebraically.
-2x+3

13. Given h( x)  1
 x3

x  2
0 x3
13. _____________________
x3
find h(-1).
1.3.3 I can write a piecewise function from a graph.
14.
14. _____________________
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1.3.5 I can identify the domain and range from a piecewise graph using interval notation.
1.3.6 I can identify and describe increasing, decreasing, and constant intervals using interval notation.
15. Determine the domain, range, and the intervals on which the function is increasing, decreasing or
Constant. Use interval notation.
16. Domain: ________________
Range: ____________________
Increasing: _________________
Decreasing: _________________
Constant: ___________________
1.3.4 I can identify whether or not a graph represents a function.
16. Is y  x  3  1 a function?
16. _____________________
Explain why or why not using complete sentences.
1.3.1 I can graph linear piecewise functions.
1
17. Graph the function: 𝑔(𝑥) = {
3
𝑥+5
|𝑥 − 3| + 1
𝑥<0
𝑥≥0
17.
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