1. Unit 1 – Chapters 1 and 4

2. Unit 1 Section 1.6-1.7 Section 4.1 Section 4.2 Section 4.3 Section 4.4
Review Ch. 4

3. Warm-Up –

4. Lesson 1.6, For use with pages 35-41
3. Gabe spent $4 more than twice as much as Casey at a store. If Casey spent $6, how much did Gabe spend?
ANSWER
$16

5. Lesson 1.7, For use with pages 42-48
1. Make a table for y = 2x + 3 with x-values of 0, 3, 6, and 9.
ANSWER x 3 6 9 y 15 21
2. Write a rule for the function.
Input, x 2 4 9
Output, y 1 7 13 28
ANSWER
y = 3x + 1

6. Vocabulary – 1.6-1.7 Domain Set of INPUTS to a function
Sometimes these are considered the X variables
AKA the “independent” variable
Range
Set of OUTPUTS of a function
Sometimes these are considered the Y variables
AKA the “dependent” variable
Quadrants
4 regions of a coordinate plane
Function
A numerical relationship where ONE input has EXACTLY ONE output
Relation
A set of inputs and corresponding outputs

7. Notes – 1.6-1.7 – Functions – Intro.
To be a Function
Each INPUT must go to EXACTLY ONE output!
Graph must pass the Vertical Line Test
Is a function a relation?
Yes
Is a relation a function?
Sometimes!!
Only if it passes the two tests above!
There are several ways to sketch graphs of equations, but the most common is THIS
GET Y BY ITSELF!!!
Build a table with at least 3 values
Sketch the graph

8. Notes – 1.6-1.7 – Functions – Intro.
4 Different Ways to view Functions
Verbal Rule – In English – Rarely used.
Graphs
Equations or Rules
Tables

9. Examples

10. EXAMPLE 1
Identify the domain and range of a function
The input-output table shows the cost of various amounts of regular unleaded gas from the same pump. Identify the domain and range of the function. 10
Input gallons
Output dollars 12 13 17
19.99
23.99
25.99
33.98
ANSWER
19.99, 23.99, 25.99, and
The domain is the set of inputs:
10, 12, 13,
and 17. The
range is the set of outputs:

11. 1 2 4 5 GUIDED PRACTICE for Example 1 Input
Identify the domain and range of the function.
Input 1 2 4
Output 5
SOLUTION
The domain is the set of inputs: 0, 1, 2, and 4 The range is the set of outputs: 1, 2, and 5

12. EXAMPLE 2
Identify a function
Tell whether the pairing is a function. a.
The pairing is not a function because the input 0 is paired with both 2 and 3.

13. EXAMPLE 2
Identify a function b.
Output
Input 2 1 4 8 6 12
The pairing is a function because each input is paired with exactly one output.

14. 2. GUIDED PRACTICE for Example 2
Tell whether the pairing is a function. 1 2
Output 12 9 6 3
Input 2.
SOLUTION
The pairing is a function because each input is paired with exactly one output.

15. 3. GUIDED PRACTICE for Example 2
Tell whether the pairing is a function. 3 2 1
Output 7 4
Input 3.
SOLUTION
2 4 7
The pairing is not a function because each input is not paired with exactly one output. IT IS A RELATION, THOUGH!!

16. EXAMPLE 3
Make a table for a function
of the function.
function, then identify the range
Make a table for the
2x is 0, 2, 5, 7, and 8.
The domain of the function y =
SOLUTION x y 2 5 7 8 = 2x 4 10 14 16
The range of the function is 0, 4, 10, 14, and 16.

17. rule for the function is y
EXAMPLE 4
Write a function rule
Write a rule for the function.
Input
Output 2 1 6 12 3 4 8 10
SOLUTION
and let y
or dependent variable. Notice that each
be the output,
Let x be the input, or independent variable,
output is 2 more than the
corresponding input.
So, a
rule for the function is y = x + 2.

18. The range of the function is 5,7,10,13 and 24.
GUIDED PRACTICE
for Examples 3,4 and 5
4. Make a table for the function y – x = 5 with domain 10, 12, 15, 18, and 29. Then identify the range of the function. HINT: GET Y BY ITSELF FIRST!!
SOLUTION x 10 12 15 18 29
y- x = 5
10 – 5 =5
12 – 5 =7
15 – 5 =10
18 – 5 =13
18 – 29 =24
The range of the function is 5,7,10,13 and 24.

19. GUIDED PRACTICE
for Examples 3,4 and 5
5. Write a rule for the function. Identify the domain and the range.
Pay (dollars) 1 2 4 3 8 16 32 24
Time (hours)
SOLUTION
Let x be the input ,or independent variable and let y be the output, or dependent variable. Notice that each output is 8 times more than corresponding input . So as a rule of function y = 8x;
domain 1, 2, 3 and 4;
range 8, 16, 24 and 32.

20. EXAMPLE 5
Write a function rule for a real-world situation
Concert Tickets
You are buying concert tickets that cost $15 each. You can buy up to 6 tickets. Write the amount (in dollars) you spend as a function of the number of tickets you buy. Identify the independent and dependent variables. Then identify the domain and the range of the function.

21. Write a function rule for a real-world situation
EXAMPLE 5
Write a function rule for a real-world situation
SOLUTION
Write a verbal model. Then write a function rule. Let n represent the number of tickets purchased and A represent the amount spent (in dollars).
Amount spent (dollars)
Cost per ticket (dollars/ticket)
Tickets purchased (tickets) = • A 15 n =
So, the function rule is A = 15n. The amount spent depends on the number of tickets bought, so n is the independent variable and A is the dependent variable.

22. EXAMPLE 5
Write a function rule for a real-world situation
Because you can buy up to 6 tickets, the domain of the function is 0, 1, 2, 3, 4, 5, and 6. Make a table to identify the range.
Amount (dollars), A 1 3 2 6 5 4 15 45 30 75 60 90
Number of tickets, n
The range of the function is 0, 15, 30, 45, 60, 75, and 90.

23. EXAMPLE 1
Graph a function
Graph the function y = x with domain 0, 2, 4, 6, and 8. 1 2
SOLUTION
STEP 1
Make an input-output table. x 2 4 6 8 y 1 3

24. EXAMPLE 1
Graph a function
STEP 2
Plot a point for each ordered pair (x, y).
WHY IS THE GRAPH A SCATTER PLOT AND NOT A LINE??

25. GUIDED PRACTICE
for Example 1
1. Graph the function y = 2x - 1 with domain 1, 2, 3, 4, and 5.
SOLUTION
STEP 1
Make an input-output table. x 1 2 3 4 5 y 7 9
STEP 2
Plot a point for each ordered pair (x, y).

26. EXAMPLE 2
Graph a function
Sat Scores
The table shows the average scores on the mathematics section of the Scholastic Aptitude Test (SAT) in the United States from 1997 to 2003 as a function of the time t in years since In the table, 0 corresponds to the year 1997, 1 corresponds to 1998, and so on. Graph the function.
519
516
514
511
512
Average score, s 6 5 4 3 2 1
Years since 1997, t

27. EXAMPLE 2
Graph a function
STEP 2
Plot the points
Does this graph pass the straight line test?
Is it a function??

28. EXAMPLE 2
Graph a function
Use the Vertical Line Test to determine if the graphs are
Functions.
Function
Function
NOT a
Function

29. Warm-Up – 4.1

30. Prerequisite Skills
SKILLS CHECK
Write the equation so that y is a function of x.
y = – x + 4 3 2
ANSWER
x + 4y = 16
ANSWER
y = – x + 1 2 5
x + 2y = 5
–12x - y = –12
ANSWER
y = -12x + 12

31. Prerequisite Skills
SKILLS CHECK
Graph the function on a coordinate plane and give the Input/ Output table.
3. y = x + 6; when x = 0, 2, 4, 6, and 8
ANSWER
Input
Output 6 2 8 4 10 12 14

32. EXAMPLE 2
Graph a function
Use the Vertical Line Test to determine if the graphs are
Functions.
Function
Function
NOT a
Function

33. Vocabulary – 4.1 Coordinate Plane
Two dimensional plane used to graph ordered pairs of numbers (x,y)
X-coordinate
The HORIZONTAL component of an ordered pair
Sometimes called the “abscissa”
Y-coordinate
The VERTICAL component of an ordered pair
Sometimes called the “ordinate”

34. Notes – 4.1 – Plot Pts. and Graphs
To plot points, move along the X axis first, and then the Y axis
You have to run before you jump (or drop!).
Domain = Inputs
Range = Outputs
Usually in Table
format
Quadrants labeled
With Roman Numerals

35. Examples 4.1

36. GUIDED PRACTICE
for Example 1
1. Use the coordinate plane in Example 1 to give the coordinates of points C, D, and E.
SOLUTION
C = (0,2)
D = (3,1)
E = (-2,-3) C.

37. GUIDED PRACTICE
for Example 1
2. What is the y-coordinate of any point on the x-axis?
SOLUTION
y-coordinate of any point on the x-axis is 0

38. EXAMPLE 3
Graph a function
Graph the function y = 2x – 1 with domain – 2, – 1, 0, 1, and 2. Then identify the range of the function.
SOLUTION
STEP 1
Make a table by substituting the domain values into the function.

39. EXAMPLE 3
Graph a function
STEP 2
List the ordered pairs: (– 2, – 5),(– 1, – 3), (0, – 1), (1, 1), (2, 3).Then graph the function.
Identify the range. The range consists of the y-values from the table: – 5, – 3, – 1, 1, and 3.

40. GUIDED PRACTICE
for Examples 2 and 3
7. Graph the function y = – x + 2 with domain – 6, –3, 0, 3, and 6. Then identify the range of the function. 1 3
STEP 1
Make a table by substituting the domain values into the function.

41. GUIDED PRACTICE
for Examples 2 and 3
y = – (6) + 2 = 0 6
y = – (3) + 2 = 1 3
y = – (0) + 2 = 2
y = – (– 3) + 2 = 3
– 3
y = – (– 6) + 2 = 4
– 6
y = – x + 2 x 1

42. GUIDED PRACTICE
for Examples 2 and 3
STEP 2
List the ordered pairs: (– 6, 4),(– 3, 3), (0, 2), (3, 1), (6,0). Then graph the function.
STEP 3
Identify the range. The range consists of the y-values from the table: 0, 1, 2, 3 and 4.

43. EXAMPLE 4
Graph a function represented by a table
VOTING
In 1920 the ratification of the 19th amendment to the United States Constitution gave women the right to vote. The table shows the number (to the nearest million) of votes cast in presidential elections both before and since women were able to vote.

44. EXAMPLE 4
Graph a function represented by a table
Years before or
since 1920
– 12
– 8
– 4 4 8 12
Votes (millions) 15 19 27 29 37 40
Explain how you know that the table represents a
function. a.
Graph the function represented by the table. b.
Describe any trend in the number of votes cast. c.

45. EXAMPLE 4
Graph a function represented by a table
SOLUTION
The table represents a function because each input has exactly one output. a.
To graph the function, let x be the number of years before or since Let y be the number of votes cast (in millions). b.
The graph of the function is shown.

46. EXAMPLE 4
Graph a function represented by a table
SOLUTION
In the three election years before 1920,the number of votes cast was less than 20 million. In 1920, the number of votes cast was greater than 20 million. The number of votes cast continued to increase in the three election years since 1920. c.

47. Warm-Up – 4.2
You may use a calculator on every assignment from this point forward unless otherwise told not to!

48. Lesson 4.2, For use with pages 215-222
1. Graph y = –x – 2 with domain –2, –1, 0, 1, and 2.
ANSWER

49. Lesson 4.2, For use with pages 215-222
Rewrite the equation so y is a function of x.
2. 3x + 4y = 16 3 4
y = – x + 4
ANSWER
3. –6x – 2y = –12
ANSWER
y = –3x + 6

50. Vocabulary – 4.2 Linear Equation
The graph of the solutions to the function form a STRAIGHT LINE!

51. Notes – 4.2 – Graph Linear Equations
Standard Form of a Linear Equation looks like this:
Ax + By = C, where A, B, and C are real numbers and A and B are not both = 0
There are several ways to sketch graphs of linear equations, but the most common is THIS
GET Y BY ITSELF!!!
Build a table with at least 3 values (negative #, positive #, and zero)
Sketch the graph
The graphs of y = constant and x = constant are special cases of linear equations.
We’ll check those out in a minute!

52. Examples 4.2

53. Standardized Test Practice
EXAMPLE 1
Standardized Test Practice
Which ordered pair is a solution of 3x – y = 7?
(3, 4) A
(1, –4) B
(5, –3) C
(–1, –2) D
SOLUTION
Check whether each ordered pair is a solution of the
equation.
Test (3, 4):
3x – y = 7
Write original equation.
3(3) – 4 = ? 7
Substitute 3 for x and 4 for y.
5 = 7
Simplify.

54. Standardized Test Practice
EXAMPLE 1
Standardized Test Practice
Test (1, – 4):
3x – y = 7
Write original equation.
3(1) – (– 4) = ? 7
Substitute 1 for x and – 4 for y.
7 = 7
Simplify.
So, (3, 4) is not a solution, but (1, – 4) is a solution of
3x – y = 7.
ANSWER
The correct answer is B. A B D C

55. EXAMPLE 2
Graph an equation
Graph the equation – 2x + y = – 3.
SOLUTION
STEP 1
Solve the equation for y.
– 2x + y = – 3
y = 2x – 3
STEP 2
Make a table by choosing a few values for x and finding the values of y. x
– 2
– 1 1 2 y
– 7
– 5
– 3

56. EXAMPLE 2
Graph an equation
STEP 3
Plot the points. Notice that the points appear to lie on a line.

57. EXAMPLE 3
Graph y = b and x = a
Graph (a) y = 2 and (b) x = – 1.
y = 2
x = -1

58. GUIDED PRACTICE
for Examples 2 and 3
Graph the equation
2. y + 3x = – 2
SOLUTION
STEP 1
Solve the equation for y.
y + 3x = – 2
y = – 3x – 2

59. GUIDED PRACTICE
for Examples 2 and 3
STEP 2
Make a table by choosing a few values for x and finding the values of y. x
– 2
– 1 1 2 y 4
– 5
– 8
STEP 3
Plot the points. Notice that the points appear to lie on a line.
STEP 4
Connect the points by drawing a line through them. Use arrows to indicate that the graph goes on without end.

60. GUIDED PRACTICE
for Examples 2 and 3
3. y = 2.5
SOLUTION
For every value of x, the value of y is 2.5. The graph of
the equation y = 2.5 is a horizontal line 2.5 units above
the x-axis.
4. x = – 4
SOLUTION
For every value of y, the value of x is – 4. The graph of
the equation x = – 4 is a vertical line 4 units to the left
of the y-axis.

61. EXAMPLE 4
Graph a linear function 1 2
Graph the function y = –
x + 4
with domain x > 0.
Then identify the range of the function.
SOLUTION
STEP 1
Make a table. x 2 4 6 8 y 3 1

62. EXAMPLE 4
Graph a linear function
STEP 2
Plot the points.
STEP 3
Connect the points with a ray because the domain is restricted.
STEP 4
Identify the range. From the graph, you can see that all points have a y-coordinate of 4 or less, so the range of the function is y ≤ 4.

63. GUIDED PRACTICE
for Example 4
5. Graph the function y = – 3x + 1 with domain x < 0. Then identify the range of the function. –
SOLUTION
STEP 1
Make a table. x
– 1
– 2
– 3
– 4 y 1 4 7 10 13

64. GUIDED PRACTICE
for Example 4
Plot the points.
STEP 2
STEP 3
Connect the points with a ray because the domain is restricted.
STEP 4
Identify the range. From the graph, you can see that all points have a y-coordinate of 1 or more, so the range of the function is y > 1. –

65. Warm-Up – 4.3

66. Daily Homework Quiz
For use after Lesson 4.2
1. Graph y + 2x = 4
ANSWER

67. Daily Homework Quiz
For use after Lesson 4.2
2. The distance in miles an elephant walks in t hours is given by d = 5t. The elephant walks for 2.5 hours. Graph the function and identify its domain and range.
domain: 0 < t < 2.5 range: 0 < d < 12.5
ANSWER –

68. Warm-up 1) Graph 5 points on the X-axis and label them.
What conclusion can you make about every point on the X-axis?
2) Graph 5 points on the Y-axis and label them.
What conclusion can you make about every point on the Y-axis?

69. Vocabulary – 4.3 X-intercept Where a graph crosses the X axis
Y-intercept
Where a graph crosses the Y axis

70. Notes – 4.3 – Graph using Intercepts.
The primary reason to use the Standard Form of a linear equation is b/c it does make finding the x and y intercepts VERY easy!
To find the X-intercept of a function
Set Y=0 and solve for X
To find the Y-intercept of a function
Set X=0 and solve for Y
Since you only need two points to make a line
Graph the X and Y intercepts and connect them!

71. Examples 4.3

72. Find the intercepts of the graph of an equation
EXAMPLE 1
Find the intercepts of the graph of an equation
Find the x-intercept and the y-intercept of the graph of 2x + 7y = 28.
SOLUTION
To find the x-intercept, substitute 0 for y and solve for x.
2x + 7y = 28
Write original equation.
2x + 7(0) = 28
Substitute 0 for y.
x =
= 14 28 2
Solve for x.

73. Find the intercepts of the graph of an equation
EXAMPLE 1
Find the intercepts of the graph of an equation
To find the y-intercept, substitute 0 for x and solve for y.
2x +7y = 28
Write original equation.
2(0) + 7y = 28
Substitute 0 for x.
y = 28 7 = 4
Solve for y.
ANSWER
The x-intercept is (14,0). The y-intercept is (0,4).

74. Find the x-intercept and the y-intercept of the graph of the equation.
GUIDED PRACTICE
for Example 1
Find the x-intercept and the y-intercept of the graph of the equation.
x + 2y = 6
SOLUTION
To find the x-intercept, substitute 0 for y and solve for x.
3x + 2y = 6
Write original equation.
3x + 2(0) = 6
Substitute 0 for y.
x = 2
Solve for x.

75. Find the intercepts of the graph of an equation for Example 1
GUIDED PRACTICE
Find the intercepts of the graph of an equation
for Example 1
To find the y-intercept, substitute 0 for x and solve for y.
3x +2y = 6
Write original equation.
3(0) + 2y = 6
Substitute 0 for x.
y =3
Solve for y.
ANSWER
The x-intercept is (2,0). The y-intercept is (0,3).

76. Use intercepts to graph an equation
EXAMPLE 2
Use intercepts to graph an equation
Graph the equation x + 2y = 4.
SOLUTION
STEP 1
Find the intercepts.
x + 2y = 4
x + 2y = 4
x + 2(0) = 4
0 + 2y = 4
x = 
x-intercept 4
y = 
y-intercept 2
Y intercept = (0,2)
X intercept = (4,0)

77. EXAMPLE 2
Use intercepts to graph an equation
STEP 2
Plot points. The x-intercept is 4, so plot the point (4, 0). The y- intercept is 2, so plot the point (0, 2). Draw a line through the points.

78. EXAMPLE 4
Solve a multi-step problem
EVENT PLANNING
You are helping to plan an awards banquet for your school, and you need to rent tables to seat 180 people. Tables come in two sizes. Small tables seat 4 people, and large tables seat 6 people. This situation can be modeled by the equation.
4x + 6y = 180
where x is the number of small tables and y is the number of large tables.
• Find the intercepts of the graph of the equation.

79. Solve a multi-step problem
EXAMPLE 4
Solve a multi-step problem
• Graph the equation.
• Give four possibilities for the number of each size table you could rent.
SOLUTION
STEP 1
Find the intercepts.
4x + 6y =
180
4x + 6y =
180
4x + 6(0) =
180
4(0) + 6y =
180
x = 
x-intercept 45
y = 
y-intercept 30

80. EXAMPLE 4
Solve a multi-step problem
STEP 2
Graph the equation.
The x-intercept is 45, so plot the point (45, 0).The y-intercept is 30, so plot the point (0, 30).
Since x and y both represent numbers of tables, neither x nor y can be negative. So, instead of drawing a line, draw the part of the line that is in Quadrant I.

81. EXAMPLE 4
Solve a multi-step problem
STEP 3
Find the number of tables. For this problem, only whole-number values of x and y make sense. You can see that the line passes through the points (0, 30),(15,20),(30, 10), and (45, 0).

82. EXAMPLE 4
Solve a multi-step problem
So, four possible combinations of tables that will seat 180 people are: 0 small and 30 large, 15 small and 20 large, 30 small and 10 large,and 45 small and 0 large.

83. 1) Do pages 10-13 from the “Classified” ads packet as a group.
Warm-Up – 4.4
1) Do pages from the “Classified” ads packet as a group.
2) You have ~15 minutes to work on this.

84. Vocabulary – 4.4 Rate of Change
Ratio of How much something changed over how long did it take to change.
Slope
The STEEPNESS of a line
Same thing as UNIT RATE!!!!
Same thing as Rate of Change!!!!
HOW FAST SOMETHING IS CHANGING!!!
Rise
Vertical or UP/DOWN change
Change in Y’s
Run
Horizontal or LEFT/RIGHT change
Change in X’s

85. Notes – 4.4–Slope and Rate of Change
Slope = RISE
RUN
4 Kinds of Slope
Positive = Slants UP
Negative = Slants DOWN
Zero = Horizontal line
No Slope = Vertical line
Finding slope with a Graph
Draw a right triangle connecting the points
Calculate RISE and RUN
Use Slope = RISE/RUN

86. Notes – Continued NOTES - CONTINUED Finding slope with a Table
Slope = Change in Y = How much change = 2ND – 1ST
Change in X = How long did it take = 2ND – 1ST
SAME as UNIT RATE and RATE OF CHANGE!!!
Pay attention to positives/negatives!!
Finding slope Using Coordinates
Variable for slope is usually m.
Slope = m = Y2 – Y1 = RISE
X2 – X1 = RUN
Plug in what you know and solve for what you don’t!

87. Examples 4.4

88. Find the slope of the line shown.
EXAMPLE 2
Find a negative slope
Find the slope of the line shown.
Let (x1, y1) = (3, 5) and (x2, y2) = (6, –1).
m =
y2 – y1
x2 – x1
Write formula for slope.
–1 – 5
6 – 3 =
Substitute.
– 6 3 = –2
Simplify.

89. Find the slope of a horizontal line
EXAMPLE 3
Find the slope of a horizontal line
Find the slope of the line shown.
Let (x1, y1) = (– 2, 4) and (x2, y2) = (4, 4).
m =
y2 – y1
x2 – x1
Write formula for slope.
4 – 4
4 – (– 2) =
Substitute. 6 =
Simplify.

90. Find the slope of a vertical line
EXAMPLE 4
Find the slope of a vertical line
Find the slope of the line shown.
Let (x1, y1) = (3, 5) and (x2, y2) = (3, 1).
m =
y2 – y1
x2 – x1
Write formula for slope.
1 – 5
3 – 3 =
Substitute.
– 4 =
Division by zero is undefined.
ANSWER
Because division by zero is undefined, the slope of a vertical line is undefined.

91. Write an equation from a graph GUIDED PRACTICE for Examples 2, 3 and 4
Find the slope of the line that passes through the points.
4. (5, 2) and (5, – 2)
SOLUTION
Let (x1, y1) = (5, 2) and (x2, y2) = (5, –2).
m =
y2 – y1
x2 – x1
Write formula for slope.
–2 – 2
5 – 5 =
Substitute.
– 4 =
Division by zero is undefined.
ANSWER
The slope is undefined.

92. EXAMPLE 5
Find a rate of change
INTERNET CAFE
The table shows the cost of using a computer at an Internet cafe for a given amount of time. Find the rate of change in cost with respect to time.
Time(hours) 2 4 6
Cost (dollars) 7 14 21

93. EXAMPLE 5
Find a rate of change
SOLUTION =
change in cost
change in time
Rate of change
14 –7
4 – 2 = 7 2
3.5
ANSWER
The rate of change in cost is $3.50 per hour.

94. GUIDED PRACTICE
for Example 5
The table shows the distance a person walks for exercise. Find the rate of change in distance with respect to time. 7.
EXERCISE
SOLUTION
Time(minute) 30 60 90
Distance (miles)
1.5 3
4.5

95. EXAMPLE 5
GUIDED PRACTICE
Find a rate of change
for Example 5 =
change in distance
change in time
Rate of change
3 – 1.5
60 – 30 =
= 0.05
ANSWER
The rate of change in cost is $0.05 mile/minute.

96. EXAMPLE 6
Use a graph to find and compare rates of change
COMMUNITY THEATER
A community theater performed a play each Saturday evening for 10 consecutive weeks. The graph shows the attendance for the performances in weeks 1, 4, 6, and 10. Describe the rates of change in attendance with respect to time.

97. Use a graph to find and compare rates of change
EXAMPLE 6
Use a graph to find and compare rates of change
SOLUTION
Find the rates of change using the slope formula.
232 – 124
4 – 1 =
108 3
Weeks 1–4:
= 36 people per week
204 – 232
6 – 4 =
– 28 2
Weeks 4–6:
= – 14 people per week
72 – 204
10 – 6 =
– 132 4
Weeks 6–10:
= – 33 people per week
ANSWER
Attendance increased during the early weeks of performing the play. Then attendance decreased, slowly at first, then more rapidly.

98. EXAMPLE 7
Interpret a graph
COMMUTING TO SCHOOL
A student commutes from home to school by walking and by riding a bus. Describe the student’s commute in words.

99. EXAMPLE 7
Interpret a graph
SOLUTION
The first segment of the graph is not very steep, so the student is not traveling very far with respect to time. The student must be walking. The second segment has a zero slope, so the student must not be moving. He or she is waiting for the bus. The last segment is steep, so the student is traveling far with respect to time. The student must be riding the bus.

100. Warm-Up – 4.5

101. Lesson 4.5, For use with pages 243-250
1. Rewrite 6x + 2y = 8 so y is a function of x.
ANSWER
y = –3x + 4
2. Find the slope of the line that passes through (–5, 6) and (0, 8). 5 2
ANSWER

102. Lesson 4.5, For use with pages 243-250
Find the intercepts of the graph of the function. (USE
THE COORDINATES OF THE POINTS!)
200x + 100y = – 600.
ANSWER
y-intercept: (0,– 6), x-intercept: (- 3,0)
4a Find the slope of -2x + y = 1.
4b Find the y-intercept of this equation as well.
HINT 1: GET Y BY ITSELF.
HINT 2: BUILD A TABLE!
ANSWER
Slope = 2 and y-intercept is 1.
Do these numbers look familiar???????

103. Vocabulary – 4.5 Parallel Lines Lines that never intersect
Lines that have the SAME SLOPE!
slope-intercept form
Linear equation where y = mx + b
m = slope of the line
b = y-intercept

104. Notes – 4.5 – Slope-Intercept Form
Slope Intercept Form of an Equation –
y = mx + b
To use the slope intercept form
Solve the equation so that Y is by itself
The coefficient of X is the slope.
The constant number is the Y intercept.
To graph a function using the slope intercept form
Graph the Y intercept
Use the slope = rise/run to find the next point
Graph the second point and connect the two points
If two lines are parallel, their slopes are ?????
Identical!
BrainPop: Slope and Intercept

105. Examples 4.5

106. Examples 4.5 – Using the Graphing Calculator
Open up the graphing application.
Graph the equation f1(x) = 2x+1 and press the Graph button. What happens?
Graph f2(x) = 2x + 3 What happens?
Press the Table button (you have to find it first!). What happens?
On the menu, press the Window/Zoom option button and choose 5- Zoom standard. What happens?
Graph the function f3(x)= x^2. Look familiar?
Clear all the functions by pressing Actions Delete All

107. EXAMPLE 1
Identify slope and y-intercept
Identify the slope and y-intercept of the line with the given equation.
y = 3x + 4 1.
3x + y = 2 2.
SOLUTION
The equation is in the form y
= m x + b.
So, the slope
of the line is 3, and the y-intercept is 4. a. b.
Rewrite the equation in slope-intercept form by solving for y.

108. Identify slope and y-intercept
EXAMPLE 1
Identify slope and y-intercept
3x + y
= 2
Write original equation. y
= – 3x + 2
Subtract 3x from each side.
ANSWER
The line has a slope of – 3 and a y-intercept of 2.

109. EXAMPLE 1
GUIDED PRACTICE
Identify slope and y-intercept
for Example 1
Identify the slope and y-intercept of the line with the given equation.
y = 5x – 3 1.
SOLUTION
The equation is in the form y
= mx + b.
So, the slope
of the line is 5, and the y-intercept is –3.

110. EXAMPLE 1
GUIDED PRACTICE
Identify slope and y-intercept
for Example 1
Identify the slope and y-intercept of the line with the given equation.
3x – 3y = 12 2.

111. Identify slope and y-intercept for Example 1
GUIDED PRACTICE
Identify slope and y-intercept
for Example 1
SOLUTION
Rewrite the equation in slope-intercept form by solving for y.
3x – 3y
= 12
Write original equation.
3x – 12
= 3y
Rewrite original equation. y
3x + 12 = 3
Divide 3 by equation. y
x – 4 =
Simplify.
ANSWER
The line has a slope of –1 and a y-intercept of –4.

112. EXAMPLE 1
GUIDED PRACTICE
Identify slope and y-intercept
for Example 1
Identify the slope and y-intercept of the line with the given equation.
x + 4y = 6 3.

113. Identify slope and y-intercept for Example 1
GUIDED PRACTICE
Identify slope and y-intercept
for Example 1
SOLUTION
Rewrite the equation in slope-intercept form by solving for y.
x + 4y
= 6
Write original equation. 4y
= – x + 6
Rewrite original equation.
– x + 6 = 4 y
Divide 3 by equation.
– x 4 + 6 =
Simplify.
ANSWER
The line has a slope of and a y-intercept of 4 1 – 2

114. EXAMPLE 2
Graph an equation using slope-intercept form
Graph the equation 2x + y = 3.
SOLUTION
STEP 1
Rewrite the equation in slope-intercept form. y
– 2x + 3 =

115. EXAMPLE 2
Graph an equation using slope-intercept form
STEP 2
Identify the slope and the y-intercept.
= – 2 m
and
= 3 b
STEP 3
Plot the point that corresponds to the y-intercept,(0, 3).
STEP 4
Use the slope to locate a second point on the line. Draw a line through the two points.

116. EXAMPLE 3
Change slopes of lines
ESCALATORS
To get from one floor to another at a library, you can take either the stairs or the escalator. You can climb stairs at a rate of 1.75 feet per second, and the escalator rises at a rate of 2 feet per second. You have to travel a vertical distance of 28 feet. The equations model the vertical distance d (in feet) you have left to travel after t seconds.
Stairs:
d = – 1.75t + 28
Escalator:
d = – 2t + 28

117. EXAMPLE 3
Change slopes of lines
a. Graph the equations in the same coordinate plane.
b. How much time do you save by taking the escalator?
SOLUTION a.
Draw the graph of d = – 1.75t + 28 using the fact that the d-intercept is 28 and the slope is – Similarly, draw the graph of d = – 2t The graphs make sense only in the first quadrant.

118. EXAMPLE 3
Change slopes of lines
The equation d = – 1.75t + 28 has a t-intercept of 16. The equation d = – 2t + 28 has a t-intercept of 14. So, you save 16 – 14 = 2 seconds by taking the escalator. b.

119. EXAMPLE 2
GUIDED PRACTICE
Graph an equation using slope-intercept form
for Examples 2 and 3
4. Graph the equation y = – 2x + 5.
SOLUTION
STEP 1
Identify the slope and the y- intercept.
= – 2 m
and
= 5 b
STEP 2
Plot the point that corresponds to the y-intercept, (0, 5).
STEP 3
Use the slope to locate a second point on the line. Draw a line through the two points.

120. EXAMPLE 5
Identify parallel lines
Determine which of the lines are
parallel.
Find the slope of each line.
– 1 – 0
– 1 – 2 =
– 1
– 3 1 3 =
Line a: m =
– 3 – (–1 )
0 – 5
– 2
– 5 = 2 5 =
Line b: m =
– 5 – (–3)
– 2 – 4
– 2
– 6 = 1 3 =
Line c: m =
ANSWER
Line a and line c have the same slope, so they are parallel.

121. EXAMPLE 5
GUIDED PRACTICE
Identify parallel lines
for Examples 4 and 5 7.
Determine which lines are parallel: line a through (-1, 2) and (3, 4); line b through (3, 4) and (5, 8); line c through (-9, -2) and (-1, 2).
SOLUTION
Find the slope of each line.
4 – 2
3 –(1) = 2
3+1 1 2 =
Line a: m =
8 – 4
5 – 3
– 4
– 2 =
Line b: m = = 2
2 – (–2)
– 1 – (– 9)
2+2
–1+ 9 = 4 8 = = 1 2
Line c: m =

122. Warm-Up – 4.6 and 4.7

123. Lesson 4.6, For use with pages 253-259
Write the equation in slope intercept form and sketch the graph
1. 4x – y = –8
ANSWER
y=4x+8
2. -9x - 3y = 21
ANSWER
y = -3x - 7

124. Lesson 4.6, For use with pages 253-259
Write the equation and sketch the graph
1. Slope = 1 and y-int = 2
ANSWER
Y = x + 2
124

125. Lesson 4.6, For use with pages 253-259
Find the unit rate.
3. You are traveling by bus. After 4.5 hours, the bus has
traveled 234 miles. Use the formula d = rt where d is
distance, r is a rate, and t is time to find the average
rate of speed of the bus.
ANSWER
52 mi/h

126. Lesson 4.6, For use with pages 253-259
Sketch the graph
1. y = 4x
ANSWER
126

127. Vocabulary – 4.6-4.7 Direct Variation Family of Functions
Linear equation where y = kx
Constant of Variation
In a Direct Variation, the letter k is the constant of variation
It’s the unit rate, the rate of change and …
SAME AS SLOPE!!
Function Notation
Different way of writing functions
F(x) means the “the function of x”
Family of Functions
A group of functions with similar characteristics (e.g. their graphs are all linear)
Parent Linear Function
Simplest form of a family of functions
F(x) = x is the parent linear function

128. Notes – 4.6-4.7 – Direct Variations and Graphing Linear Functions
A Direct variation has the form
y = kx
k = the constant of variation and is AKA
THE SLOPE!
A direct variation graph ALWAYS goes through the origin.
A direct variation is ALWAYS PROPORTIONAL!
There are two ways to find the constant, k
Find the UNIT RATE
If they give you a coordinate (x,y),
Plug in the numbers to y = kx
Solve for k.

129. Notes – 4.6-4.7 – Direct Variations and Graphing Linear Functions – cont.
A “function” is usually written as
F(x) and we read it as “F of x”
OR y = F(x)
To evaluate functions
Plug in what you know and …..???

130. Examples

131. EXAMPLE 1
Identify direct variation equations
Tell whether the equation represents direct variation. If so, identify the constant of variation.
2x – 3y = 0 a.
– x + y = 4 b.

132. Identify direct variation equations
EXAMPLE 1
Identify direct variation equations
SOLUTION
To tell whether an equation represents direct variation, try to rewrite the equation in the form y = ax.
2x – 3y = 0
Write original equation.
– 3y = – 2x
Subtract 2x from each side.
y = 2 3 x
Simplify.
ANSWER
Because the equation 2x – 3y = 0 can be rewritten in the form y = ax, it represents direct variation. The constant of variation is. 2 3

133. Identify direct variation equations
EXAMPLE 1
Identify direct variation equations b.
– x + y = 4
Write original equation.
y = x + 4
Add x to each side.
ANSWER
Because the equation – x + y = 4 cannot be rewritten in the form y = ax, it does not represent direct variation.

134. GUIDED PRACTICE
for Example 1
Tell whether the equation represents direct variation. If so, identify the constant of variation.
1. – x + y = 1

135. GUIDED PRACTICE for Example 1 SOLUTION
To tell whether an equation represents direct variation, try to rewrite the equation in the form y = ax.
– x + y = 1
Write original equation.
y = x + 1
Add x each side.
ANSWER
Because the equation – x + y = 1 cannot be rewritten in the form y = ax, it does not represent direct variation.

136. GUIDED PRACTICE for Example 1 2. 2x + y = 0 SOLUTION 2x + y = 0
Write original equation.
y = – 2x
Subtract 2x from each side.
ANSWER
Because the equation 2x + y = 0 can be rewritten in the form y = ax, it represents direct variation. The constant of variation is. – 2

137. GUIDED PRACTICE for Example 1 3. 4x – 5y = 0 SOLUTION
To tell whether an equation represents direct variation, try to rewrite the equation in the form y = ax.
4x – 5y = 0
Write original equation.
4x = 5y
Subtract Add 5y each side.
y = 4 5 x
Simplify.
ANSWER
Because the equation 4x – 5y = 0 can be rewritten in the form y = ax, it represents direct variation. The constant of variation is. 4 5

138. EXAMPLE 2
Graph direct variation equations
Graph the direct variation equation. a.
y = x 2 3
y = – 3x b.
SOLUTION a.
Plot a point at the origin. The slope is equal to the constant of variation, or Find and plot a second point, then draw a line through the points. 2 3

139. EXAMPLE 2
Graph direct variation equations
Plot a point at the origin. The slope is equal to the constant of variation, or – 3. Find and plot a second point, then draw a line through the points. b.

140. Write and use a direct variation equation
EXAMPLE 3
Write and use a direct variation equation
The graph of a direct variation equation is shown.
Write the direct variation equation. a.
Find the value of y when x = 30. b.
SOLUTION
Because y varies directly with x, the equation has the form y = ax. Use the fact that y = 2 when x = – 1 to find a. a.
y = ax
Write direct variation equation.
2 = a (– 1)
Substitute.
– 2 = a
Solve for a.

141. EXAMPLE 3
Graph direct variation equations
ANSWER
A direct variation equation that relates x and y is y = – 2x.
b. When x = 30, y = – 2(30) = – 60.

142. GUIDED PRACTICE
for Examples 2 and 3
4. Graph the direct variation equation.
y = 2x
SOLUTION
Plot a point at the origin. The slope is equal to the constant of variation, or Find and plot a second point, then draw a line through the points.

143. The graph of a direct variation on equation passes
GUIDED PRACTICE
for Examples 2 and 3
The graph of a direct variation on equation passes
through the point (4,6). Write the direct variation equation and find the value of y when x =24.
SOLUTION
Because y varies directly with x, the equation has the form y = ax. Use the fact that y = 6 when x = 4 to find a.
y = ax
Write direct variation equation.
6 = a (4)
Substitute. 6 4
a = 3 2 =
Solve for a.

144. GUIDED PRACTICE
for Examples 2 and 3
ANSWER
A direct variation equation that relates x and y is y = x when x = 24. y = (24) = 36 3 2

145. Standardized Test Practice
EXAMPLE 1
Standardized Test Practice
SOLUTION
f (x) 3x – 15 =
Write original function. =
(– 3) 3(– 3) – 15 f
Substitute -3 for x. = 24
Simplify.
ANSWER
The correct answer is A. A B C D

146. 1. Evaluate the function h(x) = – 7x when x = 7.
GUIDED PRACTICE
for Example 1
1. Evaluate the function h(x) = – 7x when x = 7.
SOLUTION
h(x) = – 7x
Write original function.
(7) = – 7(7) h
Substitute 7 for x.
= – 49
Simplify.

147. For the function f(x) 2x – 10, find the value of x so that
EXAMPLE 2
Find an x-value =
f(x)
For the function f(x) 2x – 10, find the value of x so that
= 2x – 10
f(x)
Write original function.
x – 10 =
Substitute 6 for f(x).
8 x =
Solve for x.
ANSWER
When x = 8, f(x) = 6.

148. EXAMPLE 3
Graph a function
GRAY WOLF
The gray wolf population in central Idaho was monitored over several years for a project aimed at boosting the number of wolves. The number of wolves can be modeled by the function f(x) = 37x + 7 where x is the number of years since Graph the function and identify its domain and range.

149. Review – Chap. 4

150. Daily Homework Quiz
For use after Lesson 3.8
1. Write the equation 15 = 5y – 4x so that y is a function of x
y = x + 3 4 5
ANSWER
2. Solve C = 2pr for r
ANSWER
r = p C 2
3. Solve V = Bh for B. 1 3
ANSWER
B = 3V h

151. Daily Homework Quiz
For use after Lesson 3.8
On a round-trip bicycle trip from Santa Barbara to Canada, Phil rode 2850 miles in 63 days. Find his average miles per day for the trip. Use the formula d = rt where d is distance, r is rate, and t is time.Solve for r to find the rate in miles per day to the nearest mile.
about 45 miles per day
ANSWER

152. Daily Homework Quiz
For use after Lesson 4.1
Give the coordinates of the points. Plot the points in a coordinate plane
1. and 2.
ANSWER
A(-3,2)
B(0,-1)
ANSWER

153. Daily Homework Quiz
For use after Lesson 4.1
3. A(-2,-4)
ANSWER

154. Daily Homework Quiz
For use after Lesson 4.1
4. B(3,0)
ANSWER
5. Graph y = x – 1 with domain – 4, – 2, 0, 2, 4. Then identify the range 1 2
– 3, – 2, – 1, 0, 1
ANSWER

155. Daily Homework Quiz
For use after Lesson 4.2
1. Graph y + 2x = 4
ANSWER

156. Daily Homework Quiz
For use after Lesson 4.2
2. The distance in miles an elephant walks in t hours is given by d = 5t. The elephant walks for 2.5 hours. Graph the function and identify its domain and range.
domain: 0 < t < 2.5 range: 0 < d < 12.5
ANSWER –

157. Daily Homework Quiz
For use after Lesson 4.3
1. Find the x-intercept and the y-intercept of the graph of 3x – y = 3
x-int: 1, y-int: – 3
ANSWER

158. Daily Homework Quiz
For use after Lesson 4.3
A recycling company pays $1 per used ink jet cartridge and $2 per used cartridge. The company paid a customer $14.This situation is given by x + 2y = 14 where x is the number of inkjet cartridges and y the number of laser cartridges. Use intercepts to graph the equation. Give four possibilities for the number of each type of cartridge that could have been recycled.
ANSWER
(0,7),(6,4),(10,2),(14,0)

159. Daily Homework Quiz
For use after Lesson 4.4
Find the slope of the line that passes through the points
1. (12, – 1) and (– 3, – 1)
ANSWER
2. (–2, 6) and (4, –3)
ANSWER 3 2 –

160. Daily Homework Quiz
For use after Lesson 7.2 3.
The graph shows the ticket sales for a school dance on day 1, day 3, day 6, and day 9,of ticket sales. Describe the rates of change in ticket sales with respect to time.
ANSWER
Ticket sales grew moderately, declined slightly, and then had another moderate rate of increase.

161. Daily Homework Quiz
For use after Lesson 4.5 1.
Identify the slope and y-intercept of the line 2x + 4y = –16. 1 2 –
Slope: ,y-intercept: –4
ANSWER
y = 2 3 x
ANSWER

162. Daily Homework Quiz
For use after Lesson 7.2
3. Determine which of the lines are parallel.
ANSWER
Lines a and c

163. Daily Homework Quiz
For use after Lesson 4.5 1.
Identify the slope and y-intercept of the line 2x + 4y = –16. 1 2 –
Slope: ,y-intercept: –4
ANSWER
y = 2 3 x
ANSWER

164. Daily Homework Quiz
For use after Lesson 7.2
3. Determine which of the lines are parallel.
ANSWER
Lines a and c

165. Daily Homework Quiz
For use after Lesson 4.6
Tell whether the equation represents direct variation. If so, identify the constant of variation.
x – 6y = 2
ANSWER no
x + y = 0
ANSWER
yes, – 1

166. Daily Homework Quiz
For use after Lesson 4.6 3.
The number p of parts a machine produces varies directly with the time t (in minutes) the machine is in operation. The machine produces 84 parts in 14 minutes. Write a direct variation equation that relates t and p. how many parts does the machine produce in 25 minutes?
ANSWER
p = 6t; 150 parts

167. Daily Homework Quiz
For use after Lesson 4.7
Evaluate f (x) = 8x – 4 when x = –3, 0, and 2. 1.
ANSWER
–28, –4, 12
Find the value of x so g (x) = –2x+ 1 has the value –3. 2.
ANSWER 2

168. Daily Homework Quiz
For use after Lesson 4.7
A stable charges $25 for feed and $50 per day to stable horses. The cost is given by f(x) = 50x Recently, the stable raised its fee for food to $50. The new fee is given by g(x) = 50x + 50.Graph the functions and then compare the two graphs. 3.
ANSWER
The graphs have the same slope.The y-intercept of g is 25 units greater than that of f, so g is a vertical translation.

169. Warm-Up – X.X

170. Vocabulary – X.X
Holder
Holder 2
Holder 3
Holder 4

171. Notes – X.X – LESSON TITLE.
Holder

172. Examples X.X