Rochester City School District
Functions
Unit 2
Student Workbook
2012-2013
Name: _____________________________
Functions
Lesson #1
Identifying
Mini Lesson
Define.
Relation: _______________________________________________________________________
Domain: _______________________________________________________________________
Range: ________________________________________________________________________
Function: ______________________________________________________________________
Determine if the following relations are functions.
1. Is the following relation a function? Why or why not? {(1, 5), (2, 10), (3, 15), (4, 20)}
2. Is the following relation a function? Why or why not? {4, 2), (4, 5), (3, 2), (2, 4)}
3. Is the following relation a function? Why or why not? {(4, 8), (6, 8), (8, 8)}
4. Is the following relation a function? Why or why not? {(-1, 1), (-1, 2), (1, 2), (0, 3)}
To determine if a graph is a function, we use the vertical line test. Do the graphs below show a
function? Why or why not?
1.
2.
3.
1
4. Determine if the relation is a function. y = 2x + 1
5.Determine if the relationship is a function. {(1, 4), (1, 5), (-1, 4), (-1, 5)}
6. Complete the function table and graph.
x
-2
-1
0
1
2
y = 3x - 1
y
x
-2
0
2
4
y = 1/2x + 2
y
Name: _____________________________
Lesson #1
Determine if each relation is a function. Explain.
1.
X
Y
10
-5
15
1
27
6
29
7
36
8
X
Y
2
2
6
4
4
8
0
0
2
6
2.
3.
X
Y
-6
6
-3
6
4
6
8
6
9
6
Identifying Functions
Work Period
X
Y
-6
2
-4
2
-2
4
-3
4
-4
4
X
Y
-2
2
0
4
2
6
3
6
X
Y
-5
3
2
-2
7
1
2
3
4.
5.
6.
7.
10
8
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.
8
6
4
2
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
10
9.
8
6
4
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
10.
-3
-2
3
2
1
0
-1
-1
-2
-3
-4
-5
-6
-7
-8
0
1
2
3
4
5
6
Name: _____________________________
Lesson #1 Functions
Identifying
Homework
Name: ________________________________
Lesson #2
Functions, Domain, Range
Mini Lesson
Does the following table of values represent a function? Why or why not?
x
0
2
4
2
y
4
8
12
-8
Look at the following table. Is the relation a function?
x
y
0
5
1
4
2
3
3
2
This relation is a function. Since it is a function, you can list the domain and range
Domain: {0, 1, 2, 3}
Range: {2, 3, 4, 5}
Determine if the following relation is a function, if it is state the domain and range.
1.
x
-1
0
1
2
3
x
-3
-3
-4
-4
-5
y
2
4
2
4
2
y
-5
0
5
10
15
2.
Functions can be organized in tables. The variable for the domain is called the independent variable
because it can be any number. The variable for the range is called the dependent variable because it
depends on the domain.
3. Mr. Morgan asked his students how many pets they have. Some of the student responses are shown
in the table below.
Student Number
Number of Pets
a.
1
2
3
5
6
7
Is the relation a function? Explain.
b. What is the independent and dependent variables? Explain.
c. Suppose student 7 has 2 pets. Is the relation a function? Explain.
4. In a recent 82-game season, Dwight Howard of the Orlando Magic averaged 20.7 points per game.
His approximate total point scored (p) is a function of the number of games played. (g)
a. Identify the independent and dependent variable
b. What values for the domain and range make sense for this situation?
c. Write a function (equation) to represent the total points scored. Then determine the number of
points scored in 9 games.
5. Issac belongs to a music club that charges a monthly fee of $5, plus $0.50 per song that he
downloads.
Write a function to represent the amount of money (m) he would pay in one month to download (s)
songs. What is the cost if he downloads 30 songs?
Name: ________________________________
CC 8 Lesson #2
Functions, Domain, Range
Work Period A
Identify the domain and range for each relation.
1. {(1,2) , (2,3) , (3,4) , (4,5)}
2. {(-1,3) , (-2, 5) , (-3, 7) , (-4, 9)}
Domain: ________________________
Domain: _________________________
Range: __________________________
Range: __________________________
3.
x
y
-10
1
-5
7
-3
9
0
12
Domain: ____________________________
Range: _____________________________
For the following problems, determine if each relation is a function. IF IT IS, then list the domain and
range.
4.
5. {(-6,2) , (-4,2) , (-2,4) , (-3,4) , (-4,4)}
x
y
10
-5
15
1
27
6
29
7
36
8
6.
7. {(-2,2) , (0,4) , (2,6) , (3,6)}
x
y
2
2
6
4
4
8
0
0
2
6
8.
9. {(-3,3) , (-1,5) , (0,2) , (-3,4)}
x
y
-6
6
-3
6
4
6
8
6
9
6
10. A scrapbook store is selling rubberstamps for $4.95 each. The total sale (s) is a function of the
number of rubber stamps (n) sold.
a. Identify the independent and dependent variable
b. What values for the domain and range make sense for this situation?
c. Write a function (equation) that makes sense for this situation.
11. The school bookstore rents graphing calculators for $5 per month. It also collects a non-refundable
fee of $10.00 for the school year.
a. Write the rule for the total cost (c) of renting a calculator as a function of the number of months (m).
b. If a student rented a calculator for 6 months, how much would he pay the bookstore?
Name: ________________________________
Lesson #2
Functions, Domain, Range
Workperiod B Extra practice
Name: ________________________________
Lesson #2
Functions, Domain, Range
Homework
6. A photographer takes an average of 15 pictures per session. The total number of pictures (P) is a
function of the number of sessions (s).
a. Identify the independent and dependent variable
b. What values for the domain and range make sense for this situation?
c. Write a function (equation) to represent the total number of pictures taken. Then determine the
number of pictures taken in 23 sessions.
7. Lisa is training for a marathon. She runs 85 miles per week.
a. Write an equation to find the total miles (m) run in any number of weeks (w).
b. Make a table of values to show the number of miles run in 3, 4, 5, or 10 weeks.
Name: ________________________________
Lesson #3
Linear Functions
Mini Lesson
Let us review…
What is the domain of a function?
What is the range of a function?
How are the domain and range related to the independent and dependent variable?
Complete the table of values. State the domain and range. State if the function is linear or non-linear
by graphing.
1.)
x
-4
-1
2
5
8
y = x2 – 2x
x
-3
-2
-1
0
1
y = 3x + 2
2.)
3.) The corner store sells spiral notebooks for $2.00 each and pens for $1.00 each. Tony has $10.00 to
spend. The function 𝑦 = 10 − 2𝑥 represents the number of notebooks x and pens y he can buy. Make
a table of values and graph the function.
4.) A store sells assorted nuts for $5.95 per pound. Create a table that shows the price for assorted nuts
if you purchase 1, 2, 3, or 4 pounds. Let x represent the number of pounds of nuts and y represent the
amount charged for x pounds. Is this relation a function? Explain. If yes, write a function rule for the
price of assorted nuts.
5.) Manuel is saving money for college. He already has $250. He plans to save $50 per month. Create a
table to find his total savings for 2, 4, 6, 8, and 10 months. Graph the result. Is the relation a function?
If yes, write a function rule to represent the money he is saving for college.
6.) In this problem, we will look at a table and create a function.
Marcus is planning to have his birthday at a skating rink. The rink charges a party fee plus an
additional charge for each guest.
a.
Choose two points from the table and find the rate of change.
b. Write a function to represent this situation.
c.
Number of
Guests,
x
1
2
3
4
5
6
Graph the ordered pairs
d. Use the function to find the amount the skating rink charges for the party fee.
Total
Cost $, y
53
56
59
62
65
68
Name: ________________________________
Functions: Lesson #3 Day 1
Linear Functions
Work Period
1. You are going on a road trip with your family. The speed limit is 75 miles per hour on
the freeway. Create a table of values showing how many miles you have driven over the
first 6 hours. Write an equation to represent this situation.
Rate of Change=________ Initial Value =________ Equation =______________________
2.
Courtney is collecting coins. She has 12 coins in her collection to start with and plans to
add 2 coins each week. Create a table of values showing how many coins Courtney has
in her collection over the first 8 weeks. Write an equation to model this situation.
3. Your cell phone plan costs $30 each month. For each text you send it costs $0.20.
Create a table of values showing you monthly bill if you send 0 to 80 texts. (Hint, you
may want to count by something other than 1). Write an equation to model this situation.
4. Jennifer is starting to run. She plans to run 3 miles each day. Create a table of values
showing how many total miles she has run if she continues this for 1 week. Write an
equation to model this situation.
5. Chocolate cinnamon bears cost $3.50 per pound. Create a table of values showing how
much it would cost to buy up to 8 pounds of this candy. Write an equation to model this
situation.
6. Sam gets paid $6 per hour at his new job. Sam plans to work 10 hours each week.
Create a table of values showing how much Sam will make over the first 5 weeks. Write
an equation to model this situation.
Name: ________________________________
Functions: Lesson #3 Day 2
Linear Functions
Work Period
1.) An electrician charges a base fee of $75 plus $50 for each hour of work. Create a table
that shows the amount the electrician charges for 1, 2, 3, and 4 hours of work. Let x
represent the number of hours and y represent the amount charged for x hours. Is this
relation a function? Explain. If yes, write the equation to model what an electrician
would charge.
2.) Write a relation as a set of ordered pairs in which the x-value represents the length of a
side of a square and the y-value represents the area of the square. Use a domain of 2, 4,
6, 9, and 11. Is this relation a function? Is it linear or non-linear? How do you know?
3.) The table below lists the number of grams of fat and the number of Calories from fat for
selected foods.
FOOD
Hamburger
Cheeseburger
Grilled Chicken Fillet
Breaded Chicken Fillet
Taco Salad
Grams of Fat
14
18
3.5
11
19
Calories from Fat
126
162
31.5
99
171
a.)
Create a graph for the relation between grams of fat and the
number of Calories from fat for selected foods.
b.)
Is this relation a function? Explain.
4.) You can burn about 6 Calories a minute bicycling. Let x represent the number of minutes
bicycled, and let y represent the number of Calories burned.
a.) Write ordered pairs to show the number of Calories burned if you bicycle
for 60, 120, 180, 240, or 300 minutes. Graph the ordered pairs.
b.) Find the domain and range of the relation.
c.) Does this graph represent a function? Explain.
d.) What is the unit rate identified in this situation?
e.) Write the equation to model this situation.
5.)
The table shows how much money Eric has saved. Assume the relationship between the
two quantities is linear.
a.
Find the constant rate of change and initial value
b. Write a function rule to represent this situation
c. Graph the ordered pairs
d. How much money did Eric have before he started saving?
Number
of
Months,
x
3
4
5
6
Money
Saved,
($)
y
110
130
150
170
Name: ________________________________
Lesson #4
Compare Functions
Mini Lesson
Problem 1:
Mike and Patty belong to a gym. Mike’s membership can be represented by the function c = 10m,
where c represents the cost in dollars. The cost of Patty’s membership is described in the table below.
Months
1
2
3
4
5
Cost ($)
5
10
15
20
25
a. Write a function rule to model the cost of Patty’s gym membership
b. Make a table of values to represent Mike’s gym membership
Months Cost ($)
c. Describe the rate of change for each situation
d. Who pays more for a five month membership? Explain.
Problem 2:
Cost ($)
Nina and Yelena each have a monthly cell phone bill. Yelena’s monthly bill is represented by the
function 𝑦 = 0.15𝑥 + 45, where x represents the number of minutes and y represents the cost. The
cost of Nina’s monthly cell phone bill is shown in the graph.
Minutes
Compare the y-intercepts and rate of change.
a. What is Nina’s initial cost? Yelena’s?
b. How much does Nina pay per minute to talk? Yelena?
c. What will be the monthly cost for Nina and Yelena for 200 minutes?
Name: ________________________________
Lesson #4
Compare Functions
Work Period
1. Carla’s profit at a craft fair is represented by the function 𝑝 = 5𝑏 − 15, where p is the
profit and b is the number of bracelets she sells. Ally’s profit is shown in the table.
Compare the y-intercepts and rate of change.
Bracelets Sold
1
2
3
4
a. What was Carla’s initial profit? Ally’s?
Profits ($)
5
10
15
20
b. How much does Carla charge for a bracelet? Ally?
c. How much will each girl lose or make if they sell 3 bracelets?
d. How much will each girl make if they sell 25 bracelets?
Cost ($)
2. The cost to rent an umbrella from two different companies is shown. What company
should you use if you rent an umbrella for 9 hours?
Time (h)
Pam’s
Umbrella Stand
Time (h)
Total Cost ($)
1
15.00
2
17.25
3
19.50
4
21.75
5
24.00
Name: ________________________________
Lesson #4
Compare Functions
Homework
1. For the first half of the Ramirez family trip, their speed averages 68 miles per hour. The
second half of the trip is shown in the graph below. Compare the speeds for each part of
the trip.
2. The late fees for a school library are represented by the function c = 0.25d, where c is the
total cost and d is the number of days a book is late. The fees charged by a town library
are shown in the table.
Days Late 1
2
3
a. What library charges more per day for a late fee?
Cost ($)
0.35 0.70 1.05
b. CJ checks out one book at each library and returns both books 3 days late. What are
the late fees for each library?
3. A fabric store sells cotton for $7.00 a yard. The special occasion fabric is shown in the
graph.
a. How much is special occasion fabric per yard?
b. How much is cotton fabric per yard?
c. Which fabric has a greater rate of change?
4. Andrew and Manuel purchase baseball cards each week. The amount of cards they each
have in their collection is shown in the graph and table. Who will have more cards in
week 20? Justify your answer.
Andrew’s Collection
Week Number of
Cards
1
5
2
10
3
15
4
20
Manuel’s Collection
5. Nile and Michelle played a game. Nile’s score is represented by the function 𝑝 = 5𝑐 − 3
Where p is the number of points scored and c is the correct number of answers.
Michelle’s score is shown in the table below.
Questions Answered Score
1
6
2
12
3
18
4
24
a. Compare the functions by comparing their y-intercepts and rates of change.
b. How many points will each player have it they correctly answer 21 questions?
Name: ________________________________
Lesson #5
Nonlinear Functions
Mini Lesson
Problem 1:
lbs
A
B
C
D
time
The function seen above has a dependent variable of lbs and an independent variable of time.
At time A, Jeff could initially lift 180 lbs. At time B, he increased what he could lift to 200 lbs. What may
have happened to Jeff between times B and C? What happened between times C and D? What
happened after time D?
Problem 2:
Use the table to determine whether the minimum number of Calories a tiger cub should eat is a linear
function of its age in weeks. What is the independent variable? Dependent variable? Is there a
constant rate of change?
Age
(weeks)
1
2
3
4
5
Minimum Calorie
Intake
825
1,000
1,185
1,320
1,425
Problem 3:
A cube has a side length of s meters. The volume of the cube is represented by the expression s3. Is the
volume of a cube a function? If it is, is the function linear or non-linear?
Name: ________________________________
Lesson #5
Nonlinear Functions
Work Period
1.) Write a scenario that is depicted by the graph below (you may choose to create your own
graph if you wish). You must identify the following features:
a)
b)
c)
d)
Independent and dependent variables
Initial value
Increasing and decreasing intervals of the function.
Where the function might have a maximum or minimum value.
2) The Diaz family drove from Rochester, NY to Ocean City, MD. Use the table to
determine whether the distance driven is a linear or non-linear function. Write a function
rule (equation) to model the situation.
Time (h)
Distance (miles)
1
55
2
110
3
165
4
220
3) A square has a side length of s meters. The area of the square is represented by the
expression s2. Is the area of a square a function? If it is, is the function linear or nonlinear?
4) Determine whether the table represents a linear or nonlinear function.
x 1 3
5
7
y -2 -18 -50 -98
5) Shalonda has $400 in a safe. Each month, she adds $20.00 to the safe. Manny opens a
savings account with a $400 deposit and earns 5% interest each month on the total
amount of money in the bank. Create a table of values for each situation. Explain why
one function is linear and one is not.
6) Draw a graph of a function that satisfies the following requirements:
a)
b)
c)
d)
Has an initial value that is positive.
Increases from 0 to A.
Decreases and is non-linear from A to B.
Increases with a constant rate of change from B to C.
0
A
B
C
Name: ________________________________
Lesson #5
Nonlinear Functions
Homework
The graph below shows Jack and Jill’s trip up the hill.
Jac k and Jill
1700
1600
1500
1400
1300
Distanc e (feet)
1200
1100
1000
900
800
700
600
500
400
300
200
100
0
0
2
4
6
8
10
12
14
Time (minutes)
Jill
Jack
16
18
20
22
Use the graph to answer the following questions:
1. How far from the hill is Jack at the beginning?
2. How far from the hill is Jill at the beginning?
3. How long does the trip take?
4. What is Jill doing for the first 6 minutes?
5. What happens at minute 4?
6. What is happening in the last 2 minutes?
7. Where is Jack moving fastest?
8. Where is Jill moving fastest?
9. What is the speed of Jack between 10 and 14 minutes?
10. What is the speed of Jill between 6 and 10 minutes?
11. What is Jack doing during 16 and 18 minutes?
12. What is Jill’s average speed for the whole trip?
13. What is Jack’s average speed for the whole trip?
Name: __________________________________________________________________
Comparing Linear and Nonlinear Equations
y
1) We know the graph of y  mx  b . Sketch the general appearance.
What do m and b represent? ______________________________________
What makes the graph linear?______________________________________
___________________________________________________________________
x
2) Predict which of the following five graphs will be linear.
_____________________________________________________________
What other predictions can you make about what the graphs will look like?
__________________________________________________________________________________
3) Enter the equations into the graphing calculator and sketch the graphs below.
Explain why the equation makes the graph look the way it does.
y
y
y | x |
x
y  x2
x
___________________________________
___________________________________
___________________________________
___________________________________
y
y
1
x
y
x
y
x
x
___________________________________
___________________________________
___________________________________
___________________________________
y
y  2x
___________________________________
x
___________________________________
4) Complete the table of values for the equations you graphed above. Explain how
these equations are similar to and different from the linear equation y  x
y=x
x
y = |x|
y
x
y
Similar
Different
y=
y = x2
x
y
x
Similar
Similar
Different
Different
1
x
y
y=
x
y = 2x
x
x
y
Similar
Similar
Different
Different
y
Name: ________________________________
Lesson #6
Quadratics
Work Period
1. Graph the function 𝑦 = 3𝑥 2
2. The function 𝑎 = 0.2𝑣 2 models the acceleration of a carnival ride, where a is the
acceleration toward the center of the ride in meters per second every second and v is the
velocity in meters per second. Make a table of values for velocity beginning at 0 and
ending at 10. Graph this function below. Use your graph to estimate the velocity of the
ride at an acceleration of 1 meter per second.
Name: ________________________________
Lesson #6
Quadratics
Homework
1.) A penny is dropped from a height of 196 feet off a bridge. The function 𝑦 = −16𝑥 2 + 196 models
the distance y in feet the penny is from the surface of the water at x seconds. Graph this function. Use
your graph to estimate the time it will take the penny to reach the water.
2) The area y in square feet of a projected movie on a movie screen can be represented by the
equation 𝑦 = 0.25𝑥 2 , where x represents the distance from a projector to the movie screen. Graph this
function. Use your graph to estimate the distance from the projector to a screen if the area of the
movie is 7 square feet.
Name: ____________________________________________
Please answer the following questions.
1.
2.
3.
CCSSM Function Tasks (#1)
4.
5.
Name: ____________________________________________
CCSSM Function Tasks (#2)