Unit 1

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
• Section 4.5
• Section 4.6-4.7
• Review Ch. 4

3
Warm-Up 1.6-1.7
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?
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.
x 0 3 6 9
y 3 9 15 21
ANSWER
2. Write a rule for the function.
Input, x 0 2 4 9
Output, y 1 7 13 28
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 1.6-1.7

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.



ANSWER
11
for Example 1
GUIDED PRACTICE
1. Identify the domain and range of the
function.
Input 0 1 2 4
Output 5 2 2 1
SOLUTION
The domain is the set of inputs 0, 1, 2, and
4The 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.
The pairing is a function because each input is
pairedwith exactly one output.
14
for Example 2
GUIDED PRACTICE
Tell whether the pairing is a function.
SOLUTION
The pairing is a function because each input is
pairedwith exactly one output.
15
for Example 2
GUIDED PRACTICE
Tell whether the pairing is a function.
SOLUTION
The pairing is not a function because each input
is notpaired with exactly one output. IT IS A
RELATION, THOUGH!!
16
EXAMPLE 3
Make a table for a function
SOLUTION
The range of the function is 0, 4, 10, 14, and 16.
17
EXAMPLE 4
Write a function rule
Write a rule for the function.
SOLUTION
18
for Examples 3,4 and 5
GUIDED PRACTICE
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
for Examples 3,4 and 5
GUIDED PRACTICE
5. Write a rule for the function. Identify the
domain and the range.
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
21
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)
Tickets purchased (tickets)
Cost per ticket (dollars/ticket)


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.
The range of the function is 0, 15, 30, 45, 60,
75, and 90.
23

EXAMPLE 1
Graph a function
SOLUTION
STEP 1
Make an input-output table.
x 0 2 4 6 8
y 0 1 2 3 4
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
for Example 1
GUIDED PRACTICE
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 1 3 5 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 1997.
In the table, 0 corresponds to the year 1997, 1
corresponds to 1998, and so on. Graph the
function.
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.
NOT a Function
Function
Function
29
Warm-Up 4.1
30
Prerequisite Skills
SKILLS CHECK
Write the equation so that y is a function of x.
9. 6x 4y 16
10. x 2y 5
11. 12x - y 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
Input Output
0 6
2 8
4 10
6 12
8 14
32

EXAMPLE 2
Graph a function
Use the Vertical Line Test to determine if the
graphs are Functions.
NOT a Function
Function
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
for Example 1
GUIDED PRACTICE
1. Use the coordinate plane in Example 1 to
give the coordinates of points C, D, and E.
SOLUTION
37
for Example 1
GUIDED PRACTICE
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
for Examples 2 and 3
GUIDED PRACTICE
STEP 1
Make a table by substituting the domain values
into the function.
41
for Examples 2 and 3
GUIDED PRACTICE
42
for Examples 2 and 3
GUIDED PRACTICE
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
44
EXAMPLE 4
Graph a function represented by a table
Years before or since 1920 12 8 4 0 4 8 12
Votes (millions) 15 15 19 27 29 37 40
45

EXAMPLE 4
Graph a function represented by a table
SOLUTION
46

EXAMPLE 4
Graph a function represented by a table
SOLUTION
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
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.
• Well check those out in a minute!

52
Examples 4.2

53
EXAMPLE 1
Standardized Test Practice
Which ordered pair is a solution of 3x y 7?
SOLUTION
Check whether each ordered pair is a solution of
the equation.
Test (3, 4)
Write original equation.
Substitute 3 for x and 4 for y.
Simplify.
54
EXAMPLE 1
Standardized Test Practice
Test (1, 4)
Write original equation.
Substitute 1 for x and 4 for y.
Simplify.
So, (3, 4) is not a solution, but (1, 4) is a
solution of 3x y 7.
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 0 1 2
y 7 5 3 1 1
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
for Examples 2 and 3
GUIDED PRACTICE
Graph the equation
2. y 3x 2
SOLUTION
STEP 1
Solve the equation for y.
y 3x 2
y 3x 2
59
for Examples 2 and 3
GUIDED PRACTICE
STEP 2
Make a table by choosing a few values for x and
finding the values of y.
x 2 1 0 1 2
y 4 1 2 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
for Examples 2 and 3
GUIDED PRACTICE
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
SOLUTION
STEP 1
Make a table.
x 0 2 4 6 8
y 4 3 2 1 0
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
for Example 4
GUIDED PRACTICE
SOLUTION
STEP 1
Make a table.
x 0 1 2 3 4
y 1 4 7 10 13
64
for Example 4
GUIDED PRACTICE
STEP 2
STEP 3
Connect the points with a ray because the domain
is restricted.
STEP 4
65
Warm-Up 4.3
66

Daily Homework Quiz
For use after Lesson 4.2
1. Graph y 2x 4

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.
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 Y0 and solve for X
• To find the Y-intercept of a function
• Set X0 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
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.
Write original equation.
2x 7(0) 28
Substitute 0 for y.
Solve for x.
73

EXAMPLE 1
Find the intercepts of the graph of an equation
To find the y-intercept, substitute 0 for x and
solve for y.
Write original equation.
2(0) 7y 28
Substitute 0 for x.
Solve for y.
74
for Example 1
GUIDED PRACTICE
Find the x-intercept and the y-intercept of the
graph of the equation.
1. 3x 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

EXAMPLE 1
Find the intercepts of the graph of an equation
for Example 1
GUIDED PRACTICE
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.
76
EXAMPLE 2
Use intercepts to graph an equation
Graph the equation x 2y 4.
SOLUTION
STEP 1
Find the intercepts.
x 2y 4
0 2y 4
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
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.
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
Warm-Up 4.4
1) Do pages 10-13 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 Ys
• Run
• Horizontal or LEFT/RIGHT change
• Change in Xs

85
Notes 4.4Slope and Rate of Change

• NOTES
• 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
  dont!

87
Examples 4.4

88
EXAMPLE 2
Find a negative slope
Find the slope of the line shown.
Let (x1, y1) (3, 5) and (x2, y2) (6, 1).
Write formula for slope.
Substitute.
Simplify.
89
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).
Write formula for slope.
Substitute.
Simplify.
90
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).
Write formula for slope.
Substitute.
Division by zero is undefined.
91
Write an equation from a graph
EXAMPLE 2
for Examples 2, 3 and 4
GUIDED PRACTICE
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).
Write formula for slope.
Substitute.
Division by zero is undefined.
92
EXAMPLE 5
Find a rate of change
Time(hours) 2 4 6
Cost (dollars) 7 14 21
93
EXAMPLE 5
Find a rate of change
SOLUTION
Rate of change
94
for Example 5
GUIDED PRACTICE
SOLUTION
Time(minute) 30 60 90
Distance (miles) 1.5 3 4.5
95
EXAMPLE 5
for Example 5
Find a rate of change
GUIDED PRACTICE
Rate of change
96
EXAMPLE 6
Use a graph to find and compare rates of change
97
EXAMPLE 6
Use a graph to find and compare rates of change
SOLUTION
Find the rates of change using the slope formula.
36 people per week
Weeks 14
Weeks 46
14 people per week
33 people per week
Weeks 610
98
EXAMPLE 7
Interpret a graph
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.
y 3x 4
ANSWER
2. Find the slope of the line that passes
through (5, 6) and (0, 8).
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

1. Open up the graphing application.
2. Graph the equation f1(x) 2x1 and press the
   Graph button. What happens?
3. Graph f2(x) 2x 3 What happens?
4. Press the Table button (you have to find it
   first!). What happens?
5. On the menu, press the Window/Zoom option button
   and choose 5- Zoom standard. What happens?
6. Graph the function f3(x) x2. Look familiar?
7. 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.
SOLUTION
108
EXAMPLE 1
Identify slope and y-intercept
Write original equation.
Subtract 3x from each side.
109
EXAMPLE 1
for Example 1
Identify slope and y-intercept
GUIDED PRACTICE
Identify the slope and y-intercept of the line
with the given equation.
SOLUTION
110
EXAMPLE 1
for Example 1
Identify slope and y-intercept
GUIDED PRACTICE
Identify the slope and y-intercept of the line
with the given equation.
111
EXAMPLE 1
for Example 1
Identify slope and y-intercept
GUIDED PRACTICE
SOLUTION
Rewrite the equation in slope-intercept form by
solving for y.
Write original equation.
Rewrite original equation.
Divide 3 by equation.
Simplify.
112
EXAMPLE 1
for Example 1
Identify slope and y-intercept
GUIDED PRACTICE
Identify the slope and y-intercept of the line
with the given equation.
113
EXAMPLE 1
for Example 1
Identify slope and y-intercept
GUIDED PRACTICE
SOLUTION
Rewrite the equation in slope-intercept form by
solving for y.
Write original equation.
Rewrite original equation.
Divide 3 by equation.
Simplify.
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.
115
EXAMPLE 2
Graph an equation using slope-intercept form
STEP 2
Identify the slope and the y-intercept.
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.
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
1.75. Similarly, draw the graph of d
2t 28. The graphs make sense only in the first
quadrant.
118
EXAMPLE 3
Change slopes of lines
119
EXAMPLE 2
for Examples 2 and 3
Graph an equation using slope-intercept form
GUIDED PRACTICE
4. Graph the equation y 2x 5.
SOLUTION
STEP 1
Identify the slope and the y- intercept.
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.
Line a m
Line b m
Line c m
121
EXAMPLE 5
for Examples 4 and 5
Identify parallel lines
GUIDED PRACTICE
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).
7.
SOLUTION
Line a m
Line b m
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
y4x8
ANSWER
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
Y x 2
ANSWER
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
127
Vocabulary 4.6-4.7

• Direct Variation
• Linear equation where y kx
• Constant of Variation
• In a Direct Variation, the letter k is the
  constant of variation
• Its 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 4.6-4.7

131
EXAMPLE 1
Identify direct variation equations
Tell whether the equation represents direct
variation. If so, identify the constant of
variation.
132
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.
Write original equation.
3y 2x
Subtract 2x from each side.
Simplify.
133
EXAMPLE 1
Identify direct variation equations
b.
Write original equation.
x y 4
y x 4
Add x to each side.
134
for Example 1
GUIDED PRACTICE
Tell whether the equation represents direct
variation. If so, identify the constant of
variation.
1. x y 1
135
for Example 1
GUIDED PRACTICE
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.
136
for Example 1
GUIDED PRACTICE
2. 2x y 0
SOLUTION
2x y 0
Write original equation.
y 2x
Subtract 2x from each side.
137
for Example 1
GUIDED PRACTICE
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.
Simplify.
138
EXAMPLE 2
Graph direct variation equations
Graph the direct variation equation.
SOLUTION
139
EXAMPLE 2
Graph direct variation equations
140
EXAMPLE 3
Write and use a direct variation equation
The graph of a direct variation equation is shown.
SOLUTION
y ax
Write direct variation equation.
2 a ( 1)
Substitute.
2 a
Solve for a.
141
EXAMPLE 3
Graph direct variation equations
b. When x 30, y 2(30) 60.
142
for Examples 2 and 3
GUIDED PRACTICE
4. Graph the direct variation equation.
y 2x
SOLUTION
143
for Examples 2 and 3
GUIDED PRACTICE

• 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.
Solve for a.
144
for Examples 2 and 3
GUIDED PRACTICE
145

EXAMPLE 1
Standardized Test Practice
SOLUTION
Write original function.
Substitute -3 for x.
Simplify.
146
for Example 1
GUIDED PRACTICE
1. Evaluate the function h(x) 7x when x
7.
SOLUTION
h(x) 7x
Write original function.
Substitute 7 for x.
49
Simplify.
147
EXAMPLE 2
Find an x-value
Write original function.
Substitute 6 for f(x).
Solve for x.
148


EXAMPLE 3
Graph a function
GRAY WOLF
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
2. Solve C 2pr for r

151

Daily Homework Quiz
For use after Lesson 3.8

1. 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.

152

Daily Homework Quiz
For use after Lesson 4.1
153

Daily Homework Quiz
For use after Lesson 4.1
3. A(-2,-4)
154

Daily Homework Quiz
For use after Lesson 4.1
4. B(3,0)
155

Daily Homework Quiz
For use after Lesson 4.2
1. Graph y 2x 4

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.
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
158

Daily Homework Quiz
For use after Lesson 4.3

1. 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.

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)
2. (2, 6) and (4, 3)
160
Daily Homework Quiz
For use after Lesson 7.2
161
Daily Homework Quiz
For use after Lesson 4.5
162
Daily Homework Quiz
For use after Lesson 7.2
163
Daily Homework Quiz
For use after Lesson 4.5
164
Daily Homework Quiz
For use after Lesson 7.2
165
Daily Homework Quiz
For use after Lesson 4.6
Tell whether the equation represents direct
variation. If so, identify the constant of
variation.
1. 5x 6y 2
2. x y 0
166
Daily Homework Quiz
For use after Lesson 4.6
167

Daily Homework Quiz
For use after Lesson 4.7
168

Daily Homework Quiz
For use after Lesson 4.7
169
Warm-Up X.X
170
Vocabulary X.X

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171
Notes X.X LESSON TITLE.

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172
Examples X.X