1. Functions

2. Using Graphs to Relate Two Quantities
Objective: To represent mathematical relationships using graphs.

3. Vocabulary
You can use graphs to visually represent the relationship between two variable quantities as they both change.
Tables and graphs can both show relationships between variables. Data from a table are often displayed using a graph to visually represent the relationship.
Variables are the two quantities that the graph measures.

4. Analyzing a Graph
The graph shows the volume of air in a balloon as you are blowing it up, until it pops. What are the variables? Describe how are the variables related at various points on the graph.
Air in Balloon
Volume
Time

5. Analyzing a Graph
What are the variables in each graph? Describe how the variables are related.
June Cell Phone Cost
Board Length
Cost
Length
Time
Minutes of Calls

6. Matching a Table to a Graph
A band allowed fans to download its new video from its Web site. The table shows the total number of downloads after 1, 2, 3, and 4 days. Which graph could represent the data shown in the table?
Video Downloads
Total Downloads
Total Downloads
Day
Total Downloads 1
346 2
1011 3
3455 4
10,426
Day
Day
Total Downloads
Total Downloads
Day
Day

7. Matching a Table and a Graph
The table shows the amount of sunscreen left in a can based on the number of times the sunscreen has been used. Which graph could represent the data shown in the table?
Sunscreen
Number of Uses
Amount of Sunscreen 5 1
4.8 2
4.6 3
4.4
Amount of Sunscreen
Amount of Sunscreen
Number of Uses
Number of Uses
Amount of Sunscreen
Number of Uses

8. Sketching a Graph
A model rocket rises quickly and then slows to a stop as its fuel burns out. It begins to fall quickly until the parachute opens, after which it falls slowly back to Earth. What sketch of a graph could represent the height of the rocket during its flight? Label each section.

9. Sketching a Graph
Suppose you start to swing yourself on a playground swing. You move back and forth and swing higher in the air. Then you slowly swing to a stop. What sketch of a graph could represent how your height from the ground might change over time? Label each section.

10. Practice
What are the variables in each graph? Describe how the variables are related at various points on the graph.
Total Cost
Grass Height
Amount of Paint in Can
Number of Pounds
Time
Area Painted

11. Practice
Match each graph with its related table. Explain your answers.
Time
Temp.
1 pm 91
3 pm 89
5 pm 81
7 pm 64
Time
Temp.
1 pm 61
3 pm 60
5 pm 59
7 pm 58
Time
Temp.
1 pm 24
3 pm 26
5 pm 27
7 pm 21

12. Practice
Sketch a graph to represent each situation. Label each section.
Hours of daylight each day over the course of one year
Your distance from the ground as you ride a Ferris wheel
Your pulse rate as you watch a scary movie

13. Patterns and Linear Functions
Objective: To identify and represent patterns that describe linear functions.

14. Vocabulary
The dependent variable changes in response to another variable.
The independent variable does not change. It does however cause the dependent variable to change.
Values of the independent variable are called inputs.
Values of the dependent variable are called outputs.

15. Vocabulary
A function is a relationship that pairs each input value with exactly one output value.
A linear function is a function whose graph is a nonvertical line or part of a nonvertical line.

16. Representing a Geometric Relationship
In the diagram below, what is the relationship between the number of rectangles and the perimeter of the figure they form? Represent this relationship using a table, words, an equation, and a graph. 1
Number of Rectangles, x
Perimeter, y
Ordered Pair (x,y) 1 2 3 4 6

17. Representing a Geometric Relationship
In the diagram below, what is the relationship between the number of triangles and the perimeter of the figure they form? Represent this relationship using a table, words, an equation, and a graph. 3 3 4

18. Representing a Linear Function
The table shows the relationship between the number of photos x you take and the amount of memory y in megabytes (MB) left on the camera’s memory chip. Is the relationship using words, an equation, and a graph.
Camera Memory
Number of Photos, x
Memory (MB), y
512 1
509 2
506 3
503

19. Representing a Linear Function
Is the relationship in the table below a linear function? Describe the relationship using words, an equation, and a graph.
Input, x
Output, y 8 1 10 2 12 3 14

20. Representing a Linear Function
Does the set of ordered pairs (0,2), (1,4), (3,5), and (1,8) represent a linear function? Explain.

21. Practice
For each diagram, find the relationship between the number of shapes and the perimeter of the figure they form. Represent this relationship using a table, words, an equation, and a graph. 1 1

22. Practice
For each table, determine whether the relationship is a linear function. Then represent the relationship using words, an equation, and a graph. X Y 5 1 8 2 11 3 14 X Y –3 1 2 7 3 12 X Y 43 1 32 2 21 3 10

23. Practice
For each table, determine whether the relationship is a linear function. Then represent the relationship using words, an equation, and a graph.
Gas in Tank
Mountain Climbing
Grocery Bill
Number of hours Climbing,x
Elevation, y
1127 1
1219 2
1311 3
1403
Number of Soup Cans, x
Total Bill, y
$52.07 1
$53.36 2
$54.65 3
$55.94
Miles traveled, x
Gallons of Gas, y
11.2 1
10.2 2
9.2 3
8.2

24. Patterns and Nonlinear Functions
Objective: To identify and represent patterns that describe nonlinear functions.

25. Vocabulary
A nonlinear function is a function whose graph is not a line or part of a line.
Examples of Linear Functions
Negative Slope
Positive Slope
No Slope
Examples of Nonlinear Functions
Absolute Value Cube

26. Classifying Functions as Linear or Nonlinear
The area A, in square inches, of a pizza is a function of its radius, r, in inches. The cost C, in dollars, of the sauce for a pizza is a function of the weight w, in ounces, of sauce used.
Pizza Area Sauce Cost
Radius (in.), r
Area (in.2), A 2
12.57 4
50.27 6
113.10 8
201.06 10
314.16
Weight (oz.), w
Cost, C 2
$.80 4
$1.60 8
$2.40
$3.20 10
$4.00

27. Classifying Functions as Linear or Nonlinear
The table below shows the fraction A of the original area of a piece of pater that remains after the paper has been cut in half n times. Graph the function represented by the table. Is the function linear or nonlinear?
Number of Cuts, n 1 2 3 4
Fraction of Original Area Remaining, A
1/2
1/4
1/8
1/16

28. Representing Patterns and Nonlinear Functions
The table shows the total number of blocks in each figure below as a function of the number of blocks on one edge. What is a pattern you can use to complete the table? Represent the relationship using words, and equation, and a graph.
Number of Blocks on Edge, x
Total number of Blocks, y
Ordered Pair (x,y) 1
(1,1) 2 8
(2,8) 3 27
(3,27) 4
___
(___,___) 5

29. Examples
Find the missing numbers to fill in the table. What is the pattern? What is the equation to describe the table?
Number of Figure, x
Number of Branches, y 1 3 2 9 27 4 5

30. Vocabulary
A function can be thought of as a rule that you apply to the input in order to get the output.
You can describe a nonlinear function with words or with an equation, just as you did with linear functions.

31. Writing a Rule to Describe a Nonlinear Function
The ordered pairs (1,2), (2,4), (3,8), (4,16), and (5,32) represent a function. What is a rule that represents this function?
What is a rule for the function represented by the ordered pairs (1,1), (2,4), (3,9), (4,16), and (5,25)?

32. Practice
The cost C, in dollars, for pencils is a function of the number n of pencils purchased. The length L, in inches, is a function of the time t, in seconds, it has been sharpened. Graph the function shown by each table below. Tell whether the function is linear or nonlinear.
Number of Pencils, n
Cost, C 6 $1 12 $2 18 $3 24 $4 30 $5
Time (s), t
Length (in), L
7.5 3 6 9 12
7.4 15
7.3

33. Practice
Graph the function shown by each table. Tell whether the function is linear or nonlinear. x y 5 1 2 3 x y –4 1 –3 2 3 5 x y 1 2 –5 3 8 x y 1 3 2 6 9

34. Practice
For the diagram below, the table gives the total number of small triangles y in figure number x. What pattern can you use to complete the table? Represent the relationship using words, an equation, and a graph.
Figure Number, x
Total small triangles, y
Ordered Pair (x,y) 1 3
(1,3) 2 12
(2,12) 27
(3,27) 4 5

35. Practice
Each set of ordered pairs represents a function. Write a rule that represents the function.
(0,0), (1,4), (2,16), (3,36), (4,64)
(1,2), (2,16), (3,54), (4,128), (5,250)
(1, 2 3 ), (2, 4 9 ), (3, 8 27 ), (4, ), (5, )
(0,0), (1,0.5), (2,2), (3,4.5), (4,8)

36. Graphing a Function Rule
Objective: To graph equations that represent functions.

37. Vocabulary A continuous graph is a graph that is unbroken.
A discrete graph is composed of distinct, isolated points.
The set off all solutions of an equation forms the equations graph. A graph may include solutions that do not appear in a table. A real- world graph should only show points that make sense in the given situation.

38. Graphing a Function Rule
What is the graph of the function rule y = -2x + 1?
What is the graph of the function rule y = ½ x – 1? x
y = -2x + 1
(x,y) -1
y = -2(-1) + 1
(-1,3)
y = -2(0) + 1
(0,1) 1
y = -2(1) + 1
(1,-1) 2
y = -2(2) + 1
(2,-3)

39. Graphing a Real-World Function Rule
The function rule W = 146c + 30,000 represents the total weight W, in pounds, of a concrete mixer truck that carries c cubic feet of concrete. If the capacity of the truck is about 200 cubic feet, what is a reasonable graph of the function rule?

40. Graphing a Real-World Function Rule
The function rule W = 8g represents the total weight W, in pounds, of a spa that contains g gallons of water. What is a reasonable graph of the function rule, given that the capacity of the spa is 250 gallons?
What is the weight of the spa when empty? Explain.

41. Identifying Continuous and Discrete Graphs
A local cheese maker is making cheese to sell at a farmer’s market. The amount of milk used to make the cheese and the price at which he sells the cheese are below. Write a function for each situation. Graph each function. Is the graph continuous or discrete?
1 gallon of milk makes 16 ounces of cheddar cheese.
Each wheel of cheddar cheese costs $9.

42. Identifying Continuous and Discrete Graphs
Graph each function rule. Is the graph continuous or discrete?
The amount of water w in a wading pool, in gallons, depends on the amount of time t, in minutes, the wading pool has been filling, as related by the function rule w = 3t.
The cost C for baseball tickets, in dollars, depends on the number n of tickets bought, as related by the function rule C = 16n.

43. Graphing Nonlinear Function Rule
You can also graph a nonlinear function rule. When a function rule does not represent a real world situation, graph it as a continuous function.
What is the graph of each function rule?
𝑦= 𝑥 −4
𝑦= 𝑥 2 +1
𝑦= 𝑥 3 +1

44. Writing a Function Rule
Objective: To write equations that represent functions.

45. Vocabulary
We know that one variable depends on the other variable. Once we see a pattern in a relationship, we can write a rule.
A function rule is the same as writing an equation.
Many real-world functional relationships can be represented by equations. You can use an equation to find the solution of a given real-world problem.

46. Writing a Function Rule
You can estimate the temperature by counting the number of chirps of the snowy tree cricket. The outdoor temperature is about 40°F more than one fourth the number of chirps the cricket makes in one minute. What is a function rule that represents this situation?

47. Writing a Function Rule
A landfill has 50,000 tons of waste in it. Each month it accumulates an average of 420 more tons of waste. What is a function rule that represents the total amount of waste after m months?

48. Writing and Evaluating a Function Rule
A concert seating plan is shown below. Reserved seating is sold out. Total revenue from the ticket sales will depend on the number of general-seating tickets sold. Write a function rule to represent this situation. What is the maximum possible total revenue? Reserved Seating $25 – 10 rows, 12 seats per row General Seating $10 – 30 rows, 16 seats per row

49. Writing and Evaluating a Function Rule
A kennel charges $15 per day to board dogs. Upon arrival, each dog must have a flea bath that cost $12. Write a function rule for the total cost for n days of boarding plus a bath.
How much does a 10-day stay cost?
Does a 5-day stay cost half as much as a 10-day stay?

50. Writing a Nonlinear Function Rule
Write a function rule for the area of a rectangle whose length is 5 feet more than its width. What is the area of the rectangle when its width is 9 feet?

51. Writing a Nonlinear Function Rule
Write a function rule for the area of a triangle whose height is 4 inches more than twice the length of its base. What is the area of the triangle when the base is 16 inches?

52. Practice Write a function rule to represent each situation.
The total cost C for p pounds of copper if each pound costs $3.57.
The height f, in feet, of an object when you know the object’s height h in inches.
The amount y of your friend’s allowance if the amount she receives is $2 more than the amount x you receive.
The volume V of a cube-shaped box whose edge lengths are 1 inch greater than the diameter d of the ball that the box will hold.

53. Practice Write a function rule that represents each sentence.
Y is 5 less than the product of 4 and x
C is more than half of n
7 less than three fifths of b is a
2.5 more than the quotient of h and 3 is w

54. Practice Write a function rule that represents each situatuion
A worker’s earnings e are a function of the number of hours n worked at a rate of $8.75 per hour.
The price p of a pizza is $6.95 plus $.95 for each topping t on the pizza
The load L, in pounds, of a wheelbarrow is the sum of its own 42-pound weight and the weight of the bricks that it carries. The wheelbarrow holds n 4-pound bricks.

55. Practice
A helicopter hovers 40 feet above the ground. Then the helicopter climbs at a rate of 21 feet per second. Write a rule that represents the helicopter’s height h above the ground as a function of time t. What is the helicopter’s height after 45 seconds? A team of divers assembles at an elevation of –10 feet relative to the surface of the water. Then the team dives at a rate of –50 feet per minute. Write a rule that represents the team’s depth d as a function of time t. What is the team’s depth after 3 minutes?

56. Practice
Write a function rule for the area of a triangle with a base 3 centimeters greater than 5 times its height. What is the area of the triangle when its height is 6 centimeters?
Write a functions rule for the area of a rectangle with a length 2 feet less than 3 times its width. What is the area of the rectangle when its width is 2 feet?
Write a function rule for the volume of the cylinder with a height 3 inches more than 4 times the radius of the cylinder’s base. What is the volume of the cylinder when its radius is 2 inches? (𝑉=𝜋 𝑟 2 ℎ)

57. Practice
The ordered pairs (1,2), (2,4), (3,8), (4,16), and (5,32) represent a function. What is a rule that represents this function?
What is a rule for the function represented by the ordered pairs (1,1), (2,4), (3,9), (4,16), and (5,25)?

58. Formalizing Relations and Functions
Objective: To determine whether a relation is a function. To find domain and range and use function notation.

59. Vocabulary
A relation is a paring of numbers in one set, called the domain, with numbers in another set, called the range.
A relation is often represented as a set of ordered pairs (x,y).
The domain is the set of x-values.
The range is a set of y – values.
A function is a special type of relation in which each value in the domain is paired with exactly one value in the range.

60. Identifying Functions Using Mapping Diagrams
Identify the domain and range of each relation.
(-2,5), (0,2.5), (4,6.5), (5,2.5)
(6,5), (4,3), (6,4), (5,8)
(4.2,1.5), (5,2.2), (7,4.8), (4.2,0)
(-1,1), (-2,2), (4,-4), (7, -7)

61. Vocabulary
Another way to decide if a relation is a function is to analyze the graph of the relation using the vertical line test.
If any vertical line passes through more than one point of the graph, then for some domain value there is more than one range value. So the relation is not a function.

62. Identifying Functions Using the Vertical Line Test
Is the relation a function? Use the Vertical Line Test.
(-4,2), (-3,1), (0,-2), (-4,-1), (1,2)
y = -x2 + 3
(4,2), (1,2), (0,1), (-2,2), (3,3)
(0,2), (1,-1), (-1,4), (0,-3), (2,1)

63. Vocabulary
You have seen functions represented as equations involving x and y, such as y = -3x + 1. The same equation is rewritten using function notation, f(x) = -3x + 1.
Notice that f(x) replaces y in an equation.
It is read “f of x”.
The letter f is the name of the function, not a variable.
Function notation is used to emphasize that the function value f(x) depends on the independent variable x.

64. Evaluating a Function
The function w(x) = 250x represents the number of words w(x) you can read in x minutes. How many words can you read in 8 minutes? How many words can you read in 6 minutes?

65. Finding the Range of a Function
The domain of f(x) = -1.5x + 4 is {1,2,3,4}. What is the range?
The domain of g(x) = 4x – 12 is {1,3,5,7}. What is the range?
The domain of h(x) = x2 is {-1.2,0,0.2,1.2,4}. What is the range?
The domain of f(x) = 8x – 3 is {-1/2, 1/4, 3/4, 1/8}. What is the range?

66. Identifying a Reasonable Domain and Range
You have 3 quarts of paint to paint the trim in your house. A quart of paint covers 100 squared feet. The function A(q) = 100q represents the area A(q), in square feet, that q quarts of paint cover. What domain and range are reasonable for the function? What is the graph of the function? If you have 7 quarts of paint, what are a reasonable domain and range?

67. Practice
Identify the domain and range of each relation. Use a mapping diagram to determine whether the relation is a function.
{(3,7), (3,8), (3,–2), (3,4), (3,1)}
{(6,–7), (5,–8), (1,4), (7,5)}
{(0.04,0.2), (0.2,1), (1,5), (5,25)}
{(4,2), (1,1), (0,0), (1,–1), (4,–2)}

68. Practice
Use the vertical line test to determine whether the relation is a function.
{(–2,2), (–1,1), (–1,–1), (1,1), (2,2)}
𝑦= 𝑥 2 +1
𝑦=2𝑥+1
{(0,0), (1,1), (1,–1), (2,–2), (2,2)}

69. Practice
Light travels about 186,000 miles per second. The function d(t) = t gives the distance d(t), in miles, that travels in t seconds. How far does light travel in 30 seconds?
You are buying orange juice for $4.50 per container and have a gift card worth $7. The function f(x) = 4.50x – 7 represents your total cost f(x) if you buy x containers of orange juice and use the gift card. How much do you pay to buy 4 containers of orange juice?

70. Practice Find the range of each function for the given domain.
F(x) = 2x – 7 {–2, –1, 0, 1 2}
G(x) = –4x {–5, –1, 0, 2, 10}
H(x) = 𝑥 2 {–1.2, 0, 0.2, 1.2, 4}
F(x) = 8x – 3 {− 1 2 , 1 4 , 3 4 , 1 8 }

71. Practice Find a reasonable domain and range for the function.
A car can travel 32 miles for each gallon of gasoline. The function d(x) = 32x represents the distance d(x), in miles, that the car can travel with x gallons of gasoline. The car’s fuel tank holds 17 gallons.
There are 98 International Units (IUs) of Vitamin D in 1 cup of milk. The function V(c) = 98c represents the amount V(c) of vitamin D, in IUs, you get from c cups of milk. You have a 16–cup jug of milk.

72. Sequences and Functions
Objective: To identify and extend patterns in sequences. To represent arithmetic sequences using function notation.

73. Vocabulary
A sequence is an ordered list of numbers that often form a pattern.
Each number in the list is called a term of a sequence.

74. Extending Sequences
Describe a pattern in each sequence. What are the next two terms of each sequence?
5, 8, 11, 14, __, __
2.5, 5, 10, 20, __, __
5, 11, 17, 23, __, __
400, 200, 100, 50, __, __
2, –4, 8, –16, __, __
–15, –11, –7, –3, __, __

75. Vocabulary
In an arithmetic sequence, the difference between consecutive terms is constant.
This difference is called the common difference.

76. Identifying an Arithmetic Sequence
Tell whether the sequence is arithmetic. If it is, what is the common difference? What are the next two terms?
3, 8, 13, 18
6, 9, 13, 17
8, 15, 22, 30
7, 9, 11, 13
10, 4, -2, -8
2, -2, 2, -2

77. Vocabulary
A recursive formula is a function rule that relates each term of a sequence after the first to the ones before it.
You can use the common difference of the terms of an arithmetic sequence to write a recursive formula for the sequence.
Recursive Rule: f(1) = ___ , f(n) = f(n - 1) + d
N = the term number in the sequence
F(n) = the value of the nth term of the sequence
D = the common difference
F(1) = first term

78. Writing a Recursive Formula
Write a recursive formula for the arithmetic sequence below. What is the value of the 8th term?
70, 77, 84, 91, …
3, 9, 15, 21, …
23, 35, 47, 59, …
7.3, 7.8, 8.3, 8.8, …
97, 88, 79, 70, …

79. Vocabulary
You can find the value of any term of an arithmetic sequence using a recursive formula. You can also write a sequence using an explicit formula.
An explicit formula is a function rule that relates each term of a sequence to the term number.
Rule For an Arithmetic Sequence (Explicit Rule): The nth term of an arithmetic sequence with first term A(1) and common difference d is given by A(n) = A(1) + d(n – 1)
N is the term number
A(1) is the first term
D is the common difference
A(n) is the nth term

80. Writing an Explicit Formula
An online auction works as shown below. Write an explicit formula to represent the bids as an arithmetic sequence. What is the 12th bid?
Minimum Price: $200
Bid 1: $200
Bid 2: $210
Bid 3: $220
Bid 4: $230

81. Writing an Explicit Formula
A subway pass has a starting value of $100. After one ride, the value of the pass is $ After two rides, its value is $ After three rides, its value is $ Write an explicit formula to represent the remaining value on the card as an arithmetic sequence. What is the value of the pass after 15 rides? How many rides can be taken with the $100 pass?

82. Writing an Explicit Formula From a Recursive Formula
For each recursive formula, find an explicit formula that represents the same sequence.
A(n) = A(n – 1) + 12; A(1) = 19
A(n) = A(n – 1) + 2; A(1) = 21
A(n) = A(n – 1) + 7; A(1) = 2
A(n) = A(n – 1) – 6; A(1) = 99
A(n) = A(n – 1) – 4; A(1) = 32

83. Writing a Recursive Formula From an Explicit Formula
For each explicit formula, find a recursive formula that represents the same sequence.
A(n) = (n – 1)
A(n) = (n – 1)
A(n) = 1 + 3(n – 1)
A(n) = 3 – 2(n – 1)
A(n) = 25 – 4(n – 1)

84. Practice
Describe a pattern in each sequence. Then find the next two terms of the sequence.
3, 5, 7, 9, …
9, 18, 27, 36, …
4, 12, 20, 28, …
20, 15, 10, 5, …
10, 4, –2, –8, …

85. Practice
Is the sequence arithmetic? Can you describe a pattern? What are the next two terms?
1.1, 2.2, 3.3, 4.4
19, 8, -3, -14
2, 20, 200, 2000
10, 24, 36, 52
8, 4, 2, 1
0.2, 1.5, 2.8, 4.1

86. Practice Write a recursive formula for each sequence. 13, 10, 7, 4, …
99, 88, 77, 66, …
25, 31, 37, 42, …
70, 63, 54, 37, …
32, 26, 20, 14, …

87. Practice
After one customer buys 4 new tires, a garage recycling bin has 20 tires in it. After another customer buys 4 tires, the bin has 24 tires in it. Write an explicit formula to represent the number of tires in the bin as an arithmetic sequence. How many tires are in the bin after 9 customers buy all new tires? You have a cafeteria card worth $50. After you buy lunch on Monday, its value is $ After you buy lunch on Tuesday, its value is $ Write an explicit formula to represent the amount of money left on the card as an arithmetic sequence. What is the value of the card after you buy 12 lunches?

88. Practice Write an explicit formula for each recursive formula.
A(n) = A(n – 1) + 12; A(1) = 12
A(n) = A(n – 1) +3; A(1) = 6
A(n) = A(n – 1) + 3.4; A(1) = 7.3
A(n) = A(n – 1) – 0.3; A(1) = 0.3

89. Practice Write a recursive formula for each explicit formula.
A(n) = 5 + 3(n – 1)
A(n) = –1 – 2(n – 1)
A(n) = 3 – 5(n – 1)
A(n) = 4 + 1(n – 1)

90. Practice
Find the second, fourth, and eleventh terms of the sequence described by each explicit formula.
A(n) = 5 – 3(n – 1)
A(n) = –11 + 2(n – 1)
A(n) = –3 + 5(n – 1)
A(n) = 4 + 1(n – 1)
A(n) = (n – 1)
A(n) = 1 – 6(n – 1)