Name ___________________________
Period__________
Date ___________
LF3
uc
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STUDENT PACKET
od
LINEAR FUNCTIONS
STUDENT PACKET 3: MULTIPLE REPRESENTATIONS 3
Introduction to Functions
• Define function.
• Define the graph of a function
• Explore and compare different representations of
functions.
• Determine when a set of ordered pairs is a graph of a
function.
1
LF3.2
Best Buy Problems
• Use different methods to determine the best buy, including
tables and graphs.
• Write equations that represent relationships between the
numbers of an item purchased and the cost of the
purchase.
• Define and identify direct proportion functions.
• Identify unit rates from equations and graphs.
7
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N
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LF3.1
Rate Graphs
• Solve problems involving rates, average speed, distance,
and time.
• Represent situations graphically and interpret the meaning
of specific parts of a graph.
12
LFN3.4
Vocabulary, Skill Builders, and Review
18
Sa
m
LF3.3
Linear Functions Unit (Student Packet)
LF3 – SP
Multiple Representations 3
WORD BANK
Definition or Explanation
Example or Picture
uc
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Word or Phrase
direct proportion
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function
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graph of a function
N
input-output rule
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rate
o
linear function
m
unit rate
Sa
variable
Linear Functions Unit (Student Packet)
LF3 – SP0
Multiple Representations 3
3.1 Introduction to Functions
INTRODUCTION TO FUNCTIONS
Set (Goals)
Go (Warmup)
od
• Define function.
• Define the graph of a function.
• Explore and compare different
representations of functions.
• Determine when a set of ordered pairs
is a graph of a function.
ep
r
We will explore the concept of a function.
We will define the terms function and the
graph of a function. We will describe
examples of functions and examples of
non-functions.
uc
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Ready (Summary)
N
x
x = y2 + 2
y
3
2
1
0
-1
-2
-3
Sa
m
pl
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D
x
3
2
1
0
-1
-2
-3
2.
y = x2 + 2
y
o
1.
ot
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Fill in the t-tables and draw the graphs. All points may not fit on the grids.
Linear Functions Unit (Student Packet)
LF3 – SP1
Multiple Representations 3
3.1 Introduction to Functions
WHAT IS A FUNCTION?
uc
e
A function is a rule that assigns to each input value a unique output value.
Example 1: Consider the equation y = x + 1. Here are some pairs of values that satisfy this
equation.
4
5
3
4
2
3
1
2
0
1
-1
0
-2
-1
-3
-2
od
x (input)
y (output)
Write the values in the table as ordered pairs (x, y).
2.
For the input value x = 4, can y have a value other than 5? ______
3.
Do any of the given inputs have more than one output value? _______
4.
Can you think of any input values that might have more than one output value? _____
5.
Is the rule defined by the equation a function? _______
N
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1.
7.
Name of
friend
Number
of pets
Mary
3
Kerry
1
Larry
0
Barry
0
Can two (or more) different friends have the same
number of pets? _______
Mary has 3 pets. Could Mary have exactly 3 pets
and at the same time have exactly 7 pets?_______
m
8.
Write the table as ordered pairs (input, output).
.
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6.
o
Example 2: Here is a list of 4 friends (inputs) and the number of pets they own (outputs).
Can any one friend have two different numbers of pets? _______
Sa
9.
10. Do the inputs and outputs in this table represent a function? _______
Linear Functions Unit (Student Packet)
LF3 – SP2
Multiple Representations 3
3.1 Introduction to Functions
WHAT IS A FUNCTION? (Continued)
# of people in
the apartment
(output)
od
# of bedrooms in
the apartment
(input)
1
2
3
2
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4
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1
3
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Example 3: An apartment has building nine apartments. It has two one-bedroom apartments,
four two-bedroom apartments, and three three-bedroom apartments. This mapping
diagram shows the number of bedrooms and people in the apartments.
5
6
o
N
11. Write the values in the mapping diagram as ordered pairs.
pl
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12. How many apartments have
1 bedroom? _____
2 bedrooms? _____
3 bedrooms? _____
13. How many apartments have
2 people living in it?______
3 people living in it?______
4 people living in it?______
m
1 person living in it?______
6 people living in it?______
Sa
5 people living in it?______
14. If you know the number of bedrooms in an apartment, can you determine the number of
people that live in that apartment?_____
15. Does this mapping diagram represent a function? _____
Linear Functions Unit (Student Packet)
LF3 – SP3
Multiple Representations 3
3.1 Introduction to Functions
PRACTICE WITH FUNCTIONS
y
4
6
8
6
4
x
0
2
4
6
8
y
10
9
8
7
6
d.
x
1
2
3
4
5
y
9
9
9
9
9
x
9
9
9
9
9
y
1
2
3
4
5
ep
r
x
0
3
6
9
12
c.
od
b.
a.
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1. Which of the following input-output tables could illustrate functions if the variable x is used
for the input value and y for the output value? ________________
5. Which of the following sets of ordered pairs could illustrate functions? ______________
(10, 5), (10, 6), (10, 7), (10, 8)
c.
(0, 4), (1, 4), (2, 4), (3, 4)
b.
(1, 5), (2, 6), (3, 5), (4, 6)
d.
(10, -20), (-20, 10), (-10, -5), (10, 5)
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a.
a.
1
b.
2
8
3
6
6
5
c.
-1
-3
3
m
1
-5
d.
1
-1
-3
3
5
-5
8
Sa
5
1
8
4
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5
2
4
o
3
N
6. Which of the following mapping diagrams could illustrate functions? ________________
7. Choose one example from above that is not a function and explain why.
Linear Functions Unit (Student Packet)
LF3 – SP4
Multiple Representations 3
3.1 Introduction to Functions
THE GRAPH OF A FUNCTION
uc
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The graph of a function is its set of ordered pairs (x, y). For any point on the graph of a
function, its x-coordinate will represent an input value and its y-coordinate will represent its
corresponding output value. If the inputs and outputs are real numbers, then we can
represent the graph of a function as points on the coordinate plane.
od
The vertical line test is a way to determine whether a set of ordered pairs in the usual xy-plane
is the graph of a function. The vertical line test simply states that if a vertical line intersects a
graph in more than one point, then it is a NOT a function.
b.
d.
Sa
m
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c.
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a.
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1. Use the vertical line test. Which of the following graphs could represent functions?
2. Try the vertical line test on the graphs you created in the warmup. Do these graphs
represent functions? Explain.
Linear Functions Unit (Student Packet)
LF3 – SP5
Multiple Representations 3
3.1 Introduction to Functions
DRAWING GRAPHS
uc
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Draw graphs to fit each description.
2. A nonlinear function
(a function whose graph is not a line)
4. A nonlinear “non-function”
(a graph that is not a line, and does not
represent a function)
m
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3. A linear “non-function”
(a graph that is a line, and does not
represent a function)
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od
1. A linear function
(a function whose graph is a line)
Sa
5. Explain why your graphs for problems 1 and 2 represent functions.
6. Explain why your graphs for problems 3 and 4 do not represent functions.
Linear Functions Unit (Student Packet)
LF3 – SP6
Multiple Representations 3
3.2 Best Buy Problems
BEST BUY PROBLEMS
Set (Goals)
od
• Use different methods to determine the
best buy, including tables and graphs.
• Write equations that represent
relationships between the numbers of
an item purchased and the cost of the
purchase.
• Define and identify direct proportion
functions.
• Identify unit rates from equations and
graphs.
ot
R
ep
r
We will use numbers and graphs to help
determine which choices are better buys.
We will learn about a special linear function
called a direct proportion.
uc
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Ready (Summary)
Go (Warmup)
VALUE-MART
N
Suppose you are running out of your favorite pens and pencils, so you compare prices at two
stores before making a purchase.
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D
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Pens: 6 for $7.50
Pencils: 12 for $6.80
SAVINGS HUT
Pens: 6 for $8.25
Pencils: 14 for $6.80
Sa
m
1. At which store are pens cheaper? Explain.
2. At which store are pencils cheaper? Explain.
Linear Functions Unit (Student Packet)
LF3 – SP7
Multiple Representations 3
3.2 Best Buy Problems
SHMEAR ‘N THINGS
4 bagels for $3.00
HOLE-Y BREAD
5 bagels for $4.00
5. Label and scale the grid. Graph the data
using two different colors.
1. Complete the tables. Assume a
proportional relationship.
5
8
10
12
15
16
20
20
25
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4
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HOLE-Y
BREAD
# of
cost
bagels
(y)
(x)
ot
R
SHMEAR ‘N
THINGS
# of
cost
bagels
(y)
(x)
uc
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BAGELS
N
2. Which is the better buy? Use entries in
the tables to explain your reasoning.
o
6. Explain which graph illustrates a slower
rise in price.
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3. Write equations to relate the number of
bagels to cost.
SHMEAR ‘N THINGS
y = ________________
HOLE-Y BREAD
y = ________________
m
The linear functions you wrote above are
both in the form y = mx. This is called a
direct proportion equation because y is
directly proportional to (is a multiple of) x.
7. Identify the coordinates when x = 1
SHMEAR ‘N THINGS
(1, ____)
HOLE-Y BREAD
(1, ____)
8. What does this represent in the context of
the problem?
Sa
4. How is this equation different from the
linear function y = mx + b?
Linear Functions Unit (Student Packet)
LF3 – SP8
Multiple Representations 3
3.2 Best Buy Problems
FLAT ‘N ROUND
3 tortillas for $0.60
WRAP IT UP
4 tortillas for $1.00
6. Label and scale the grid. Graph the data
using two different colors.
1. Complete the tables. Assume a
proportional relationship
# of
tortillas
(x)
4
6
8
cost
(y)
ot
R
3
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WRAP IT UP
ep
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FLAT ‘N
ROUND
# of
cost
tortillas
(y)
(x)
uc
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TORTILLAS
N
2. Which is the better buy? Use entries in
the tables to explain your reasoning.
o
7. Explain which graph illustrates a slower
rise in price.
pl
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3. Write equations to relate the number of
tortillas to cost.
FLAT ‘N ROUND
y = ___________________
WRAP IT UP
y = ___________________
4. Identify the coordinates when x = 1
m
FLAT ‘N ROUND
WRAP IT UP
(1, ____)
(1, ____)
Sa
5. How are these coordinates related to the
unit rate for one tortilla?
Linear Functions Unit (Student Packet)
In the linear function y = mx + b,
b represents the y-intercept.
8. Write coordinates for the y-intercepts for
each function.
FLAT ‘N ROUND
(0, ____)
WRAP IT UP
(0, ____)
9. What does this represent in the context of
the problem?
LF3 – SP9
Multiple Representations 3
3.2 Best Buy Problems
PAPA’S PITA
6 pitas for $____
EAT-A PITA
10 pitas for $____
uc
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PITA BREAD
# of
pitas
(x)
cost
(y)
2
$1.00
EAT-A PITA
# of
pitas
(x)
cost
(y)
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$5
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PAPA’S PITA
od
1. Complete the tables and graphs. The graph for EAT-A PITA is provided. A partial table for
PAPA’s PITA is provided. Use tables and graphs to complete pricing circles above.
Assume proportional relationships.
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2. Which is the better buy? Use entries in the
tables or graphs to explain your
reasoning.
3. Write equations to relate the number of
pitas to cost.
y = ______________
EAT-A PITA
y = ______________
m
PAPA’S PITA
Sa
4. How can you determine unit rate from the
equation?
Linear Functions Unit (Student Packet)
$0
1
5
5. Identify the coordinates when x = 1
PAPA’S PITA
(1, ____)
EAT-A PITA
(1, ____)
6. What does this represent in the context of
the problem?
7. Identify the coordinates when x = 0
PAPA’S PITA
(0, ____)
EAT-A PITA
(0, ____)
What does this represent in the context of
the problem?
LF3 – SP10
Multiple Representations 3
3.2 Best Buy Problems
CROISSANTS
CURVEY’S
8 croissants for $____
uc
e
MOON’S
5 croissants for $____
# of
croissants
(x)
CURVEY’S
cost
(y)
# of
croissants
(x)
cost
(y)
4
$2.00
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$5
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MOON’S
od
1. Complete the tables and graphs. The graph for Moon’s Croissants is provided. A partial
table for Curvey’s is provided. Use tables and graphs to complete pricing circles above.
$0
N
2. Which is the better buy? Use entries in
the tables or graphs to explain your
reasoning.
1
5
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5. Identify the coordinates when x = 1
3. Write equations to relate the number of
croissants to cost.
MOON’S
y = _____________
CURVEY’S
y = _____________
Sa
m
4. How can you determine unit rate from the
equation?
Linear Functions Unit (Student Packet)
MOON’S
(1, ____)
CURVEY’S
(1, ____)
6. What does this represent in the context of
the problem?
7. Identify the coordinates when x = 0
MOON’S
(0, ____)
CURVEY’S
(0, ____)
What does this represent in the context
of the problem?
LF3 – SP11
Multiple Representations 3
3.3 Rate Graphs
RATE GRAPHS
Set (Goals)
uc
e
Ready (Summary)
ep
r
Go (Warmup)
od
• Solve problems involving rates, average
speed, distance, and time.
• Represent situations graphically and
interpret the meaning of specific parts of
a graph.
We will use words, pictures, tables of
numbers, and graphs to represent rates.
We will relate the various representations
to each other.
Chris went jogging at the park. Use the graph to complete the table.
Distance
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600 yd
Number of
Minutes
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Time Period
o
0
N
200 yd
From 0 minutes to
2 minutes
2.
From 2 minutes to
4 minutes
3.
From 0 minutes to
4 minutes
Distance
Traveled
4 min
Graph is not drawn to scale.
Average
Rate of Speed
m
1.
2 min
Time
Sa
4. In what part of the jog did Chris run faster, the initial two minutes or the last two
minutes? Explain by referencing numbers and the shape of the graph.
Linear Functions Unit (Student Packet)
LF3 – SP12
Multiple Representations 3
3.3 Rate Graphs
POURING WATER 1
uc
e
Your teacher will give you a small cup and a clear container. Fill up the small cup with water
and pour it into the clear container. After each pour, you will measure and record the height of
the water in millimeters.
2. Make a graph of the data in your table.
o
N
Height in Millimeters
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1. Make a sketch of the clear
container used
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3. Record your data in the table
Number of
Pours
1
Height in
Millimeters
2
3
m
4
Sa
5
6
7
Linear Functions Unit (Student Packet)
0
Number of Pours
LF3 – SP13
Multiple Representations 3
3.3 Rate Graphs
POURING WATER 2
2. Container 2
Graph: ______
Graph: ______
Explain:
Explain:
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od
1. Container 1
uc
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Suppose you poured water into these containers at a constant rate.
• Match each container with an appropriate graph below.
• Write one or two sentences to justify each choice.
4. Container 4
Explain:
Explain:
C.
D.
Height of Water
m
Sa
Number of Pours
Height of Water
B.
Height of Water
A.
Graph: ______
Height of Water
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Graph: ______
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N
3. Container 3
Number of Pours
Linear Functions Unit (Student Packet)
Number of Pours
Number of Pours
LF3 – SP14
Multiple Representations 3
3.3 Rate Graphs
Suppose you poured water into these different containers at a constant rate.
• Sketch a graph for each.
• Write one or two sentences to justify each sketch.
6. Container 6
Graph:
Graph:
Height of Water
o
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Height of Water
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5. Container 5
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POURING WATER 2 (Continued)
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Number of Pours
Explain:
Sa
m
Explain:
Number of Pours
Linear Functions Unit (Student Packet)
LF3 – SP15
Multiple Representations 3
3.3 Rate Graphs
MATCH THE TABLE TO THE GRAPH
uc
e
Without plotting points, match each input-output table with a graph below. Write one or two
sentences to justify each choice. Look for constant, increasing, or decreasing rates of change.
2.
1.
Input (x)
0
1
2
3
4
5
6
Output (y)
1
4
7
10
13
16
19
od
Output (y)
1
3
5
7
9
11
13
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Input (x)
0
1
2
3
4
5
6
Graph: ______
Explain:
3.
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Graph: ______
Explain:
4.
o
B.
Explain:
C.
y
D.
y
y
Sa
m
y
Output (y)
1
7
12
16
19
21
22
Graph: ______
Graph: ______
Explain:
A.
Input (x)
0
1
2
3
4
5
6
N
Output (y)
1
2
4
7
11
16
22
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Input (x)
0
1
2
3
4
5
6
x
Linear Functions Unit (Student Packet)
x
x
x
LF3 – SP16
Multiple Representations 3
3.3 Rate Graphs
MAKE THE NUMBERS FIT
uc
e
Make up an appropriate table of numbers for each graph. Use estimates only.
1.
y
x
y
od
5
x
ep
r
5
2.
x
y
y
3.
y
x
60
y
30
4.
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o
x
N
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50
y
y
30
Sa
m
x
x
30
Linear Functions Unit (Student Packet)
x
30
LF3 – SP17
Multiple Representations 3
3.4 Vocabulary, Skill Builders, and Review
VOCABULARY, SKILL BUILDERS, AND REVIEW
Match the words to the clues.
Words
Clues
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FOCUS ON VOCABULARY
a. This graph is a straight line. It is
expressed in the form y = mx + b
1. __________graph of a function
b. A rate for one unit of measure.
Example: 80 miles per hour
ot
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2. __________input-output rule
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od
1. __________function
c.
A rule that establishes an output value
for each given input value
d. A ratio in which the numbers have
units attached to them.
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3. __________linear function
N
Example: y = 6x – 2
e. A linear function where one variable is a
multiple of another
5. __________unit rate
f.
m
4. __________rate
Sa
6. __________ variables
7. __________direct proportion
Linear Functions Unit (Student Packet)
In the equation d = rt, the quantities d,
r, and t are _________.
g. Ordered pairs represented on a
coordinate grid
h. The set of ordered pairs (x, y)
LF3 – SP18
Multiple Representations 3
3.4 Vocabulary, Skill Builders, and Review
SKILL BUILDER 1
Compute.
-3 + (-5)
3.
4.
8−3
5.
6 + (-6)
6.
7.
-3 − (-12)
8.
-6 − 8
10.
-8 • 2
13.
14 ÷ (-7)
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9.
10
-2
o
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14.
-32
-4
9 − (-4)
5 − (-8)
12.
-24
6
15.
5 • (-3)
N
11.
4−7
uc
e
2.
od
-9 + 2
ep
r
1.
Solve.
17. If 6 pencils cost $1.38, then what is the
cost of 1 pencil?
Sa
m
16. If 1 pencil costs $0.19, then what is the
cost of 5 pencils?
Linear Functions Unit (Student Packet)
LF3 – SP19
Multiple Representations 3
3.4 Vocabulary, Skill Builders, and Review
SKILL BUILDER 2
1. -6 + x < -4
Think (write in words):
Think (write in words):
Solution:
Solution:
Graph:
Graph:
Check a point on the ray.
Check a point on the ray.
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od
2. -x > -5 + 3
uc
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Solve each inequality using a mental strategy and graph.
Solve each equation using a mental strategy or cups and markers.
3. 10 – b = -3
k−2
-6
= -3
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e:
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o
N
4.
6. 30 = -5(m + 2)
7. 2x + 3 = x + 5
8. 3 (x – 4 ) = 12
Sa
m
5. -8 = 8 + n
Linear Functions Unit (Student Packet)
LF3 – SP20
Multiple Representations 3
3.4 Vocabulary, Skill Builders, and Review
SKILL BUILDER 3
1.
Step 3
Step #
1
2
# of
toothpicks
3
5
Step 4
3
4
5
od
Step 2
. . . 10
. . . 250
ep
r
Step 1
uc
e
For each problem, sketch step 4, complete the table, define your variables, and write an
explicit rule for the number of toothpicks based on the step number.
ot
R
Define variables:
Rule using symbols:
N
2.
1
Step 3
2
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D
Step #
Step 2
o
Step 1
3
Step 4
4
5
. . . 10
. . . 250
# of toothpicks
Define variables:
m
Rule using symbols:
Sa
Translate each word phrase into symbols.
3. The product of a number and 5,
decreased by 7
Linear Functions Unit (Student Packet)
4. The difference between 7 and the product
of a number and 5
LF3 – SP21
Multiple Representations 3
3.4 Vocabulary, Skill Builders, and Review
SKILL BUILDER 4
od
y
1
7
9
5
7
4. (1, -2), (-2, 1), (-1, -2), (2, 1)
6.
2
8
2
8
4
6
4
6
1
3
5
N
1
3
5
9
3. (0, 2), (1, -2), (2, 2), (3, -2)
x
3
5
1
3
9
ep
r
5.
2.
y
1
3
5
7
9
ot
R
1.
x
1
3
5
7
9
uc
e
For each of the following tables, sets of ordered pairs, mapping diagrams, or graphs write YES
if it could represent a function or NO if it could not. For the NOs, explain why not.
8.
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D
o
7.
10
Sa
m
9. Make up reasonable x-y values for the graph. Could this graph represent a function?
Explain.
y
x
y
10
Linear Functions Unit (Student Packet)
x
LF3 – SP22
Multiple Representations 3
3.4 Vocabulary, Skill Builders, and Review
SKILL BUILDER 5
Cost
(y)
# of
pairs
(x)
cost
(y)
ot
R
ep
r
# of
pairs
(x)
HOSIERY HUT
od
SOCKS ‘R WE
HOSIERY HUT
6 pairs of socks for $7.80
uc
e
SOCKS ‘R WE
4 pairs of socks for $6.00
1. Fill in the table. Which is the better buy?
Use entries in the tables to explain your
reasoning.
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2. Label and scale the grid. Graph the data
using two different colors. Explain which
graph illustrates a slower rise in price.
3. Find the unit rates for pairs of socks at both shops. Use these numbers to explain which
has the better buy.
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4. Write equations to relate the number of pairs of socks to cost.
y = ______________________
HOSIERY HUT
y = ______________________
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SOCKS ‘R WE
5. How can you determine unit rate from the equation?
Linear Functions Unit (Student Packet)
LF3 – SP23
Multiple Representations 3
3.4 Vocabulary, Skill Builders, and Review
SKILL BUILDER 6
Match each equation to one set of ordered pairs and also to one graph.
Ordered pairs
y = 2x
2.
y = 2x + 3
3.
y = 3x + 2
4.
y = 3x – 2
5.
y = -x + 3
6.
y = -x
a.
(0, 0) (1, -1) (-1, 1)
c.
(0, 3) (1, 5) (-1, 1)
e.
(0, 2) (1, 5) (-1, -1)
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h.
(0, 0) (1, 2) (-1, -2)
d.
(0, 3) (1, 2) (-1, 4)
f.
(0, -2) (1, 1) (-1, -5)
i.
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g.
b.
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od
1.
Graph
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Equation
k.
l.
Sa
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j.
Linear Functions Unit (Student Packet)
LF3 – SP24
Multiple Representations 3
3.4 Vocabulary, Skill Builders, and Review
SKILL BUILDER 7
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Pat went running at the park. Use the graph to complete the table.
od
400 yd
0
1 min
4 min
Time
Graph is not drawn to scale.
Distance traveled
Average rate of speed
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Time period
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Distance
600 yd
From 0 minutes to 1 minute
2.
From 1 minute to 4 minutes
3.
From 0 minutes to 4 minutes
N
1.
5. Suppose you poured water into this
container at a constant rate. Sketch a graph
and write one or two sentences to justify it.
Sa
m
Explain:
Linear Functions Unit (Student Packet)
Graph:
Height of Water
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4. In what part of the jog did Pat run faster, the initial one minute or the last three minutes?
Explain by referencing numbers and the shape of the graph.
Number of Pours
LF3 – SP25
Multiple Representations 3
3.4 Vocabulary, Skill Builders, and Review
SKILL BUILDER 8
1. Graph: ______ Explain:
y
B.
Output (y)
19
16
13
10
7
4
1
od
Input (x)
0
1
2
3
4
5
6
y
C.
y
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A.
Output (y)
13
11
9
7
5
3
1
ep
r
Input (x)
0
1
2
3
4
5
6
2. Graph: ______ Explain:
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Without plotting the ordered pairs, match each input-output table with a graph below.
Write one or two sentences to justify each choice.
x
x
N
x
Make up an appropriate table of numbers for each graph. Use estimates only.
3.
4.
o
y
x
y
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x
y
y
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x
Linear Functions Unit (Student Packet)
x
LF3 – SP26
Multiple Representations 3
3.4 Vocabulary, Skill Builders, and Review
SKILL BUILDER 9
2.
B
m∠A =
50 ,
C
m∠B =
90 ,
D
od
A
E
m∠C =
_____
F
m∠D =
120 , m∠E = m∠F , m∠E =
____
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1.
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Apply each geometry fact to find angles.
The sum of the measures of the angles in a triangle = 180o.
If two angles are supplementary, then their sum is 180o.
4.
p
m∠p =
140 ,
q
m∠q =
_____
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3.
w x
m∠w = m∠x
m∠w =
______
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6. Solve the problem. Use guess and check if needed.
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Guess
o
Two angles are supplementary. One angle is
58o more than the other. Find the larger
angle.
First angle
Make and sketch
and
Check
S = _____?
Sum of angles
Second angle
High? Low? Correct?
m
x
Equation
Sa
Write an equation and solve it if you can.
Linear Functions Unit (Student Packet)
and
Solution
Answer the question.
Verify the solution using the equation.
LF3 – SP27
Multiple Representations 3
3.4 Vocabulary, Skill Builders, and Review
TEST PREPARATION
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Show your work on a separate sheet of paper and choose the best answer.
1. Which graph best matches the input-output below?
A. y
1
2
B.
2
4
3
6
4
8
5
10
od
0
0
C. y
y
x
x
x
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x
D. y
ep
r
Input (x)
Output (y)
2. The Office Supply Store and Office Plus both sell notebooks. The Office Supply Store
sells 8 notebooks for $7.12. Office Plus sells 5 notebooks for $5.25. Which store offers
the better buy?
N
The Office Supply Store
# of
8 16 24
notebooks (x)
cost (y)
40
o
32
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A. Office Plus
Office Plus
# of
notebooks (x)
cost (y)
10 20 30 40 50
B. Office Supply
C. The prices are the same
D. Can’t tell from information given.
3. Which representation below could match the linear function graphed here?
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A. The table
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Input (x)
Output (y)
B. The ordered pairs
0
0
2
-1
-2
1
-3
2
C. The equation
y = -2x + 0
Linear Functions Unit (Student Packet)
(-1, 2)
(2, -3)
(1, -2)
(-2, 3)
D. The equation
y =
2
x
1
LF3 – SP28
Multiple Representations 3
3.4 Vocabulary, Skill Builders, and Review
KNOWLEDGE CHECK
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Show your work on a separate sheet of paper and write your answers on this page.
3.1 Introduction to Functions
Which of the following could represent a function? Explain.
2. y = 4x – 5
3.
4.
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y
1
2
3
4
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x
0
1
1
2
od
1. (2, 5) (3, 5) (4, 5) (5, 5)
3.2 Best Buy Problems
T-Shirt Mania and Shirts R’ Us sell souvenir t-shirts. T-Shirt Mania charges $18 for
three t-shirts and Shirts R’ Us charges $25 for four t-shirts.
N
5. Find the unit rates for t-shirts at both stores. Use the numbers to explain which store has
the better buy.
o
6. Write the equations to relate the number of t-shirts to cost for both stores.
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T-Shirt Mania
# of t-shirts (x)
3
6
cost (y)
9
12
Shirts R’ Us
# of t-shirts (x)
4
8
12
16
cost (y)
3.3 Rate Graphs
m
7. Make an appropriate table of numbers for each graph. Use estimates only.
y
y
5
Sa
x
x
Linear Functions Unit (Student Packet)
5
LF3 – SP29
Multiple Representations 3
Sa
m
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This page is left intentionally blank for notes.
Linear Functions Unit (Student Packet)
LF3 – SP30
Multiple Representations 3
Sa
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This page is left intentionally blank for notes.
Linear Functions Unit (Student Packet)
LF3 – SP31
Multiple Representations 3
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This page is left intentionally blank for notes.
Linear Functions Unit (Student Packet)
LF3 – SP32
Multiple Representations 3
HOME-SCHOOL CONNECTION
Here are some questions to review with your young mathematician.
8
16
24
32
40
# of
cookies (x)
cost (y)
10
20
30
40
50
ep
r
# of
cookies (x)
cost (y)
Cookieland
od
Cookies n’ Things
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1. Cookies n’ Things charges $3.20 for 8 cookies. Cookieland charges $4.50 for 10
cookies. Which store has the better buy for cookies?
x
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2. Make an appropriate table of numbers for each graph. Use estimates only.
Q
y
N
5
x
5
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3. Could the graph from problem 2 represent a function? Explain.
Sa
Parent (or Guardian) Signature ____________________________
Linear Functions Unit (Student Packet)
LF3 – SP33
Multiple Representations 3
COMMON CORE STATE STANDARDS – MATHEMATICS
STANDARDS FOR MATHEMATICAL CONTENT
7.RP.2b
7.RP.2c
7.RP.2d
8.EE.5
8.F.1
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Compare properties of two functions each represented in a different way (algebraically, graphically, numerically
in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a
linear function represented by an algebraic expression, determine which function has the greater rate of
change.
Construct a function to model a linear relationship between two quantities. Determine the rate of change and
initial value of the function from a description of a relationship or from two (x, y) values, including reading these
from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the
situation it models, and in terms of its graph or a table of values.
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the
function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of
a function that has been described verbally.
Create equations in two or more variables to represent relationships between quantities; graph equations on
coordinate axes with labels and scales.
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8.F.4
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8.F.2
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7.RP.1
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6.EE.9
Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in
the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
Use variables to represent two quantities in a real-world problem that change in relationship to one another;
write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity,
thought of as the independent variable. Analyze the relationship between the dependent and independent
variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion
at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to
represent the relationship between distance and time.
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities
measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit
rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions
of proportional relationships.
Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of
items purchased at a constant price p, the relationship between the total cost and the number of items can be
expressed as t = pn.
Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with
special attention to the points (0, 0) and (1, r) where r is the unit rate.
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different
proportional relationships represented in different ways. For example, compare a distance-time graph to a
distance-time equation to determine which of two moving objects has greater speed.
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the
set of ordered pairs consisting of an input and the corresponding output.
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6.RP.3a
8.F.5
A-CED-2
STANDARDS FOR MATHEMATICAL PRACTICE
Make sense of problems and persevere in solving them.
MP2
Reason abstractly and quantitatively.
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MP1
Construct viable arguments and critique the reasoning of others.
MP4
Model with mathematics.
MP5
Use appropriate tools strategically.
MP6
Attend to precision.
MP7
Look for and make use of structure.
MP8
Look for and express regularity in repeated reasoning.
Sa
MP3
Linear Functions Unit (Student Packet)
LF3 – SP34