Lesson 1.1 Skills Practice
Name
1
Date
A Picture Is Worth a Thousand Words
Understanding Quantities and Their Relationships
Vocabulary
Write a definition for each term in your own words.
1. independent quantity
The quantity that the dependent quantity depends on is the independent quantity.
2. dependent quantity
When one quantity depends on another in a problem situation, it is said to be the
dependent quantity.
Problem Set
Determine the independent and dependent quantities in each scenario.
1. Selena is driving to visit her grandmother who lives 325 miles away from Selena’s home. She travels
an average of 60 miles per hour.
Independent quantity: time (hours)
Dependent quantity: distance (miles)
2. Benjamin works at a printing company. He is making T-shirts for a high school volleyball team.
The press he runs can imprint 3 T-shirts per minute with the school’s mascot.
© 2012 Carnegie Learning
Independent quantity: time (minutes)
Dependent quantity: number of T-shirts imprinted
3. On her way to work each morning, Sophia purchases a small cup of coffee for $4.25 from the
coffee shop.
Independent quantity: number of cups
Dependent quantity: cost (dollars)
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4. Phillip enjoys rock climbing on the weekends. At some of the less challenging locations he can climb
upwards of 12 feet per minute.
Independent quantity: time (minutes)
Dependent quantity: distance climbed (feet)
5. Jose prefers to walk to work when the weather is nice. He walks the 1.5 miles to work at a speed of
about 3 miles per hour.
Independent quantity: time (hours)
Dependent quantity: distance (miles)
6. Gavin works for a skydiving company. Customers pay $200 per jump to skydive in tandem skydives
with Gavin.
Independent quantity: number of jumps
Dependent quantity: cost (dollars)
Choose the graph that best models each scenario.
7. Kylie is filling her backyard pool to get ready for the summer. She is using a garden hose to fill the
pool at a rate of 14 gallons per minute.
Graph A
y
0
Graph B
y
x
0
Graph C
y
x
0
x
8. Hector is training to participate in competitive trampoline. In his best jump, he can reach a maximum
height of about 9 meters and can spend about 2 seconds in the air performing tricks.
Graph A
y
0
Graph B
y
x
0
Graph C
y
x
0
© 2012 Carnegie Learning
Graph A
x
Graph C
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Name
Date
9. Jasmine is saving for college. She has invested $500 in a mutual fund that is expected to earn an
average of 7% annually.
Graph A
Graph B
y
0
Graph C
y
x
0
y
x
x
0
Graph B
10. Each day Maria starts her walk to school at 7:45 am. At 7:50 am she stops at her friend Jenna’s house.
Jenna is usually late and Maria must wait at least 5 minutes for her to get ready. At 7:55 am Maria and
Jenna leave Jenna’s house and arrive at school at 8:10 am.
Graph A
Graph B
© 2012 Carnegie Learning
y
0
Graph C
y
y
x
0
x
0
x
Graph B
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11. Marcus is at the top of an observation tower. He drops an action figure with a parachute attached
and watches it descend to the ground.
Graph A
Graph B
y
y
0
Graph C
x
0
y
x
0
x
Graph C
12. Janelle holds a raffle to raise money for a children’s hospital. Participants who enter the raffle guess
the number of peanuts in a jar. Janelle records the number of peanuts each participant guesses and
the number of peanuts their guess is off by.
Graph B
y
0
y
y
x
Graph A
254 Graph C
0
x
0
x
© 2012 Carnegie Learning
Graph A
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Lesson 1.1 Skills Practice
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Name
Date
Label the axes of the graph that models each scenario with the independent and dependent quantities.
13. Madison enjoys bicycling for exercise. Each Saturday she bikes a course she has mapped out around
her town. She averages a speed of 12 miles per hour on her journey.
Distance Madison Bikes
Distance (miles)
y
0
x
Time (hours)
14. Natasha is filling the bathtub with water in order to give her dog Buster a bath. The faucet fills the tub
at an average rate of 12 gallons per minute.
Amount of Water in Bathtub
Volume (gallons)
© 2012 Carnegie Learning
y
0
x
Time (minutes)
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15. Marcus throws a football straight up into the air. After it reaches its maximum height of 20 feet, it
descends back to the ground.
Football Height
Height of Football (feet)
y
0
x
Time (seconds)
16. Chloe is using a pump to drain her backyard pool to get ready for winter. The pump removes the
water at an average rate of 15 gallons per minute.
0
256 x
Time (minutes)
© 2012 Carnegie Learning
Amount of Water in Pool
Volume (gallons)
y
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Date
17. Jermaine is saving money to purchase a used car. He places $850 dollars in a savings account that
earns 1.65% interest annually.
Interest Earned
Interest (dollars)
y
0
x
Time (years)
18. Zachary enjoys hiking. On the first day of his latest hiking trip, he hikes through flat terrain for about
8 miles. On the second day, he hikes through very steep terrain for about 3 miles. On the third day he
hikes through some hilly terrain for about 6 miles.
Distance Hiked
Distance (miles)
© 2012 Carnegie Learning
y
0
x
Time (days)
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Lesson 1.2 Skills Practice
1
Name
Date
A Sort of Sorts
Analyzing and Sorting Graphs
Vocabulary
© 2012 Carnegie Learning
Match each definition to its corresponding term.
1. A graph with no breaks in it
b. continuous graph
a. discrete graph
2. The mapping between a set of inputs and a set of outputs
c. relation
b. continuous graph
3. The set of all input values of a relation
e. domain
c. relation
4. The set of all output values of a relation
f. range
d. function
5. A graph of isolated points
a. discrete graph
e. domain
6. A visual method used to determine whether a relation represented as
a graph is a function
g. Vertical Line Test
f. range
7. A relation between a given set of elements for which each input value
there exists exactly one output value
d. function
g. Vertical Line Test
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Problem Set
Each pair of graphs has been grouped together. Provide a rationale to explain why these graphs may have
been grouped together.
1. Graph A Graph B
y
y
8
8
6
6
4
4
2
2
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22
4
6
8
x
28 26 24 22
0 2
22
24
24
26
26
28
4
6
8
x
28
Answers will vary.
Both graphs are always decreasing from left to right. Both graphs are functions. Both graphs are
made up of straight lines.
2. Graph A Graph B
y
8
8
6
6
4
4
2
2
28 26 24 22
0 2
22
4
6
8
x
0 2
28 26 24 22
22
24
24
26
26
28
4
6
8
x
28
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y
Answers will vary.
Both graphs are smooth curves. Both graphs are functions. Both graphs have an increasing and
a decreasing interval. Both graphs are U-shaped.
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Lesson 1.2 Skills Practice
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Name
Date
3. Graph A Graph B
y
y
8
8
6
6
4
4
2
2
0 2
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22
4
6
8
x
0 2
28 26 24 22
22
24
24
26
26
28
4
6
8
x
28
Answers will vary.
Both graphs have an increasing and a decreasing interval. Both graphs have a minimum value.
Both graphs are functions.
4. Graph A Graph B
© 2012 Carnegie Learning
y
y
8
8
6
6
4
4
2
2
0 2
28 26 24 22
22
4
6
8
x
0 2
28 26 24 22
22
24
24
26
26
28
4
6
8
x
28
Answers will vary.
Both graphs have discrete data values. Both graphs are decreasing from left to right. Both graphs
are functions.
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5. Graph A Graph B
y
y
8
8
6
6
4
4
2
2
0 2
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22
4
6
8
x
28 26 24 22
0 2
22
24
24
26
26
28
4
6
8
4
6
8
x
28
Answers will vary.
Both graphs are increasing from left to right. Both graphs are functions.
6. Graph A Graph B
y
y
8
6
6
4
4
2
2
0 2
28 26 24 22
22
4
6
8
x
0 2
28 26 24 22
22
24
24
26
26
28
x
28
Answers will vary.
Both graphs have an increasing and a decreasing interval. Both graphs have a maximum value.
Both graphs are functions.
262 © 2012 Carnegie Learning
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Lesson 1.2 Skills Practice
1
page 5
Name
Date
Determine whether the graph is discrete or continuous.
7. 8.
9.
y
y
y
x
The graph is discrete.
The graph is continuous.
10. 11.
The graph is continuous.
12.
y
y
y
x
x
x
© 2012 Carnegie Learning
x
x
The graph is continuous.
The graph is discrete.
The graph is continuous.
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Lesson 1.2 Skills Practice
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Determine if each graph represents a function by using the Vertical Line Test.
13. 14.
y
y
8
8
6
6
4
4
2
2
0 2
28 26 24 22
22
4
6
8
x
0 2
28 26 24 22
22
24
24
26
26
4
6
8
x
28
28
Yes. The graph is a function.
No. The graph is not a function.
15. 16.
y
8
8
6
6
4
4
2
2
0 2
28 26 24 22
22
4
6
8
x
0 2
28 26 24 22
22
24
24
26
26
4
6
28
28
No. The graph is not a function. Yes. The graph is a function.
264 8
x
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y
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Lesson 1.2 Skills Practice
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Name
Date
17. 18.
y
y
8
8
6
6
4
4
2
2
0 2
28 26 24 22
22
4
6
8
x
0 2
28 26 24 22
22
24
24
26
26
4
6
8
x
28
28
© 2012 Carnegie Learning
No. The graph is not a function. Yes. The graph is a function.
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Lesson 1.3 Skills Practice
1
Name
Date
There Are Many Ways to Represent Functions
Recognizing Algebraic and Graphical Representations of Functions
Vocabulary
Choose the term from the box that best completes each statement.
function notation
function family
absolute maximum
linear piecewise functions
1.
increasing function
linear functions
quadratic functions
decreasing function
exponential functions
linear absolute value functions
constant function
absolute minimum
Function notation
is a way to represent equations algebraically that makes it more
efficient to recognize the independent and dependent variables.
exponential functions
2. The family of
includes functions of the form f(x) 5 a ? bx, where a
and b are real numbers, and b is greater than 0 but is not equal to 1.
linear piecewise functions
3. The family of
for different parts, or pieces, of the domain.
includes functions that have an equation that changes
4. When both the independent and dependent variables of a function increase across the entire domain,
increasing function
the function is called an
.
absolute maximum
5. A function has an
if there is a point on its graph that has a
y-coordinate that is greater than the y-coordinates of every other point on the graph.
6. A
function family
is a group of functions that share certain characteristics.
© 2012 Carnegie Learning
7. The family of linear absolute value functions includes functions of the form f(x) 5 a​| x 1 b |​1 c,
where a, b, and c are real numbers, and a is not equal to 0.
8. When the dependent variable of a function decreases as the independent variable increases across
decreasing function
.
the entire domain, the function is called a
quadratic functions
9. The family of
includes functions of the form f(x) 5 ax2 1 bx 1 c,
where a, b, and c are real numbers, and a is not equal to 0.
linear functions
includes functions of the form f(x) 5 ax 1 b, where a
10. The family of
and b are real numbers, and a is not equal to 0.
11. If the dependent variable of a function does not change or remains constant over the entire domain,
constant function
.
then the function is called a
absolute minimum
12. A function has an
if there is a point on its graph that has a
y-coordinate that is less than the y-coordinates of every other point on the graph.
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Lesson 1.3 Skills Practice
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Problem Set
Rewrite each function using function notation.
1. Rewrite the function y 5 3x 2 8 using function notation so that the dependent quantity, defined as f,
is a function of the independent quantity x.
f(x) 5 3x 2 8
2. Rewrite the function y 5 3​x2​ ​1 6x 21 using function notation so that the dependent quantity, defined
as C, is a function of the independent quantity x.
C(x) 5 3​x​2​1 6x 2 1
3. Rewrite the function y 5 ​3x​​1 8 using function notation so that the dependent quantity, defined as P,
is a function of the independent quantity x.
P(x) 5 ​3x​​1 8
4. Rewrite the function l 5 |n 2 2| using function notation so that the dependent quantity, defined as L,
is a function of the independent quantity n.
L(n) 5 |n 2 2|
1 ​  m 1 5 using function notation so that the dependent quantity, defined as
5. Rewrite the function d 5 2​ __
2
A, is a function of the independent quantity m.
A(m) 5 2​ 1  ​ m 1 5
2
6. Rewrite the function c 5 2p r2 using function notation so that the dependent quantity, defined as C,
is a function of the independent quantity r.
C(r) 5 2p r2
268 © 2012 Carnegie Learning
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Lesson 1.3 Skills Practice
1
page 3
Name
Date
Choose the graph that represents each function. Use your graphing calculator.
7. f(x) 5 __
​ 2 ​  x 1 2
3
Graph A
Graph B
Graph C
y
y
y
x
x
x
Graph A
8. f(x) 5 2x2 1 4
Graph A
Graph B
y
Graph C
y
x
y
x
x
© 2012 Carnegie Learning
Graph C
9. f(x) 5 2x 1 5
Graph A
Graph B
Graph C
y
y
y
x
x
x
Graph B
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Lesson 1.3 Skills Practice
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10. f(x) 5 |x 2 6|
Graph A
Graph B
Graph C
y
y
y
x
x
x
Graph C
11. f(x) 5 2x 2 6, where x is an integer
Graph A
Graph B
y
Graph C
y
x
y
x
x
Graph A
12. f(x) 5 24
Graph B
y
Graph C
y
y
x
x
x
© 2012 Carnegie Learning
Graph A
Graph B
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Lesson 1.3 Skills Practice
1
page 5
Name
Date
Determine whether each graph represents an increasing function, a decreasing function, a constant
function, or a combination of increasing and decreasing functions.
13. 14.
15.
y
y
x
y
x
x
The graph represents an The graph represents an
increasing function.
increasing function.
16. 17.
18.
y
y
x
© 2012 Carnegie Learning
The graph represents a
function with a combination
of an increasing interval and
a decreasing interval.
y
x
The graph represents a
decreasing function.
x
The graph represents a
constant function.
The graph represents a
decreasing function.
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Lesson 1.3 Skills Practice
1
page 6
Determine whether each graph represents a function with an absolute minimum, an absolute maximum,
or neither.
20.
21.
y
y
y
x
The graph represents a
function with an absolute
minimum.
22. The graph represents a function with neither an absolute minimum nor an absolute maximum.
23.
The graph represents a
function with an absolute
maximum.
24.
y
y
y
x
x
The graph represents a function with neither an absolute minimum nor an absolute maximum.
272 x
x
x
The graph represents a
function with an absolute
maximum.
The graph represents a
function with an absolute
minimum.
© 2012 Carnegie Learning
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Lesson 1.3 Skills Practice
1
page 7
Name
Date
Determine whether each graph represents a linear function, a quadratic function, an exponential function,
a linear absolute value function, a linear piecewise function, or a constant function.
25. 26.
27.
y
y
x
y
x
The graph represents an exponential function.
28. The graph represents a
linear function.
29.
The graph represents a linear
piecewise function.
30.
y
y
y
x
© 2012 Carnegie Learning
x
x
x
The graph represents a quadratic function.
The graph represents a
constant function.
The graph represents a linear
absolute value function.
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Lesson 1.4 Skills Practice
1
Name
Date
Function Families for 200, Alex…
Recognizing Functions by Characteristics
Problem Set
Choose the appropriate function family or families to complete each sentence based on the given
characteristic(s).
linear functions
quadratic functions
exponential functions linear absolute value functions
1. The graph of this function family is a straight line. The function family is
linear functions
.
2. The graph of this function family has an increasing interval and a decreasing interval. The function
family is quadratic functions or linear absolute value functions .
3. The graph of this function family has an absolute minimum. The function family is
quadratic functions or linear absolute value functions .
4. The graph of this function family in decreasing over the entire domain. The function family is
linear functions or exponential functions .
5. The graph of this function family forms a V shape. The function family is
value functions
.
linear absolute
© 2012 Carnegie Learning
6. The graph of this function family has an increasing interval and a decreasing interval and forms
quadratic functions
a U shape. The function family is
.
7. The graph of this function family does not have an absolute maximum or absolute minimum and
exponential functions
is a smooth curve. The function family is
.
8. The graph of this function family has an absolute maximum or absolute minimum and is made up
linear absolute value functions
straight lines. The function family is
.
9. The graph of this function family is made up straight lines and does not have an absolute maximum
linear functions
or absolute minimum. The function family is
.
10. The graph of this function family decreases over the entire domain and is a smooth curve.
exponential functions
The function family is
.
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Lesson 1.4 Skills Practice
1
page 2
Create an equation and sketch a graph for a function with each set of given characteristics. Use values
that are any real numbers between 210 and 10.
11. Create an equation and sketch a graph that:
• is a smooth curve,
• is continuous,
• has a minimum, and
• is quadratic.
Answers will vary.
y
f(x) 5 x
2
8
6
4
2
0 2
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22
4
6
8
4
6
8
x
24
26
28
12. Create an equation and sketch a graph that:
• is linear,
• is discrete, and
• is decreasing across the entire domain.
Answers will vary.
8
6
4
2
0 2
28 26 24 22
22
24
x
© 2012 Carnegie Learning
f(x) 5 2x, where x is an integer
y
26
28
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Lesson 1.4 Skills Practice
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Name
Date
13. Create an equation and sketch a graph that:
• is a smooth curve,
• is increasing across the entire domain,
• is continuous, and
• is exponential.
Answers will vary.
y
f(x) 5 2
x
8
6
4
2
28 26 24 22
0 2
22
4
6
8
4
6
8
x
24
26
28
14. Create an equation and sketch a graph that:
• has a maximum,
• is continuous, and
• is a linear absolute value function.
Answers will vary.
© 2012 Carnegie Learning
f(x) 5 |x|
y
8
6
4
2
0 2
28 26 24 22
22
x
24
26
28
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Lesson 1.4 Skills Practice
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15. Create an equation and sketch a graph that:
• is linear,
• is continuous,
• is neither increasing nor decreasing across the entire domain, and
• does not pass through the origin.
Answers will vary.
y
f(x) 5 3
8
6
4
2
0 2
28 26 24 22
22
4
6
8
4
6
8
x
24
26
28
16. Create an equation and sketch a graph that:
• is discrete,
• has a maximum,
• does not pass through the origin, and
• is quadratic.
Answers will vary.
y
f(x) 5 2x2 1 3, where x is an integer
8
4
0
28 26 24
22
24
26
x
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6
28
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Lesson 1.4 Skills Practice
1
page 5
Name
Date
Choose the function family represented by each graph.
linear function
linear absolute value function
quadratic function
linear piecewise function
17. exponential function
18.
y
y
8
8
6
6
4
4
2
2
0 2
28 26 24 22
22
4
6
8
x
0 2
28 26 24 22
22
24
24
26
26
28
4
6
8
x
28
The graph represents a quadratic function.
19. The graph represents a linear function.
20.
© 2012 Carnegie Learning
y
y
8
8
6
6
4
4
2
2
0 2
28 26 24 22
22
4
6
8
x
0 2
28 26 24 22
22
24
24
26
26
28
4
6
8
x
28
The graph represents a linear The graph represents an exponential
absolute value function. function.
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Lesson 1.4 Skills Practice
1
21. page 6
22.
y
y
8
8
6
6
4
4
2
2
0 2
28 26 24 22
22
4
6
8
x
0 2
28 26 24 22
22
24
24
26
26
4
6
8
x
28
28
© 2012 Carnegie Learning
The graph represents a linear The graph represents a linear function.
piecewise function.
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