Unit 1

1
Unit 1 Chapter 5
2
Unit 1

• Section 5.1 Write Linear Equations in
  Slope-Intercept Form
• Section 5.2
• Section 5.3
• Section 5.4
• Section 5.5
• Section 5.6
• Section 5.7
• Chapter 5 Review

3
Warm-Up 5.1
4
Lesson 5.1, For use with pages 282-291
Find the slope of the line that passes through
the points.
1. (2, 1), (4, 0)
ANSWER
2. (1, 3), (1, 5)
ANSWER
4
5
Lesson 5.1, For use with pages 282-291
3. A landscape architect charges 75 for a
consulting fee and 30 per hour. Write an
equation that shows the cost C as a function of
time t (in hours).
ANSWER
C 30t 75
6
Lesson 5.1, For use with pages 282-291
Evaluate
1. f(x) - x 2, x 2
ANSWER
2. f(x) 2x-3. Find x if f(x) 11
ANSWER
x 7 because f(7) 2(7) 3 11
7
Vocabulary 5.1

• Y-intercept
• Where graph crosses the y-axis
• Where the story starts
• Slope
• How fast something changes!
• AKA
• Unit rate, steepness, rate of change, constant of
  variation, etc.
• Slope-Intercept Form
• Ymxb
• Standard Form
• Ax By C

8
Notes 5.1 Write LE in Slope-Int Form.

• Slope Intercept form
• Y Mx b
• Need TWO things to write an equation in Slope-Int
  form. There are many ways to get them!
• Slope
• rise/run, y2-y1/x2-x1, how much over how long,
  draw triangles on graphs, etc.
• Y - Intercept
• Read graph, set x0 and solve for y, find b in
  slope-intercept form

9
Examples 5.1

10
EXAMPLE 1
Use slope and y-intercept to write an equation
Write an equation of the line with a slope of 2
and a y -intercept of 5.
y mx b
Write slope-intercept form.
y 2x 5
Substitute 2 for m and for b.
11
EXAMPLE 2
Standardized Test Practice
Which equation represents the line shown?
The line crosses the y-axis at (0, 3). So, the
y-intercept is 3.
y mx b
Write slope-intercept form.
12
EXAMPLE 2
Standardized Test Practice
13
for Examples 1 and 2
GUIDED PRACTICE
Write an equation of the line with the given slop
and y-intercept.
1. Slope is 8 y-intercept is 7.
SOLUTION
y mx b
Write slope-intercept form.
Substitute 8 for m and 7 for b.
y 8x 7
14
for Examples 1 and 2
GUIDED PRACTICE
SOLUTION
y m x b
Write slope-intercept form.
15
EXAMPLE 3
Write an equation of a line given two points
Write an equation of the line shown.
16
EXAMPLE 3
Write an equation of a line given two points
SOLUTION
STEP 1
Calculate the slope.
Write an equation of the line. The line crosses
the y-axis at (0, 5). So, the y-intercept is
5.
STEP 2
y mx b
Write slope-intercept form.
17
Write a linear function
EXAMPLE 4
Write an equation for the linear function f with
the values f(0) 5 and f(4) 17.
SOLUTION
STEP 1
Write f(0) 5 as (0, 5) and f (4) 17 as (4,
17).
Calculate the slope of the line that passes
through (0, 5) and (4, 17).
STEP 2
18

Write a linear function
EXAMPLE 4
STEP 3
Write an equation of the line. The line crosses
the y-axis at (0, 5). So, the y-intercept is 5.
y mx b
Write slope-intercept form.
y 3x 5
Substitute 3 for m and 5 for b.
19
GUIDED PRACTICE
for Examples 3 and 4
3. Write an equation of the line shown.
SOLUTION
STEP 1
Calculate the slope.
STEP 2
Write an equation of the line. The line comes the
y-axis at (0, 1). So, the y-intercept is 1.
y mx b
Write slope-intercept form.
20
GUIDED PRACTICE
for Examples 3 and 4
f(0) 2, f(8) 4
SOLUTION
STEP 1
Write f(0) 2 as (0, 2 ) and f (8) 4 as (8,
4).
STEP 2
Calculate the slope of the line that passes
through (0, 2) and (8, 4).
21
GUIDED PRACTICE
for Examples 3 and 4
STEP 3
Write an equation of the line. The line comes the
y-axis at (0, 2). So, the y-intercept is 2.
y mx b
Write slope-intercept form.
22
Warm-Up 5.2
23
Lesson 5.2, For use with pages 292-299
Find the equation of the line that passes through
the points.
1. (0, 2), (1, 3)
ANSWER
Y 5x -2
2. (3, 2), (0, 2)
ANSWER
Y 4/3 x - 2
24
Lesson 5.2, For use with pages 282-291
3. Bill wants a video game that costs 75. After
borrowing 75 from his parents, he is paying
back 5 per week. Write an equation that tells
how much money Bill owes after x weeks.
ANSWER
y 5x 75

• Using the slope-intercept form, find b if
• slope 2, x 2 and y 9. Then find the
  equation of the line.

ANSWER
b 5 Equation y 2x 5
25
Vocabulary 5.2

• Slope-Intercept Form
• Y mx b
• Y-intercept
• Where a graph crosses the y axis
• X 0
• X-intercept
• Where a graph crosses the x axis
• Y0

26
Notes 5.2 Use Linear Eqns in Slope-Int. Form.

• To use slope intercept form, I need two pieces
  of information
• Slope
• Y-intercept
• If you have ONE point and the slope, you can find
  the y-intercept by doing the dance!
• Plug in the slope and (x,y) to the slope-int form
  and solve for b.
• To find out if 3 points are on the same line,
• Find the lin. Eqn. for the first two points.
• Plug in the (x,y) coordinates from the third
  point to the eqn. and see if it works.

27
Examples 5.2

28
EXAMPLE 1
Write an equation given the slope and a point
Write an equation of the line that passes through
the point ( 1, 3) and has a slope of 4.
SOLUTION
STEP 1
Identify the slope. The slope is 4.
Find the y-intercept. Substitute the slope and
the coordinates of the given point in y m x
b. Solve for b.
STEP 2
y m x b
Write slope-intercept form.
3 4( 1) b
Substitute 4 for m, 1 for x, and 3 for y.
29
EXAMPLE 1
Write an equation given the slope and a point
1 b
Solve for b.
STEP 3
Write an equation of the line.
y m x b
Write slope-intercept form.
y 4x 1
Substitute 4 for m and 1 for b.
30
for Example 1
GUIDED PRACTICE
Write an equation of the line that passes through
the point (6, 3) and has a slope of 2.
SOLUTION
STEP 1
Identify the slope. The slope is 2.
Find the y-intercept. Substitute the slope and
the coordinates of the given point in y m x
b. Solve for b.
STEP 2
y m x b
Write slope-intercept form.
3 2(6) b
Substitute 3 for y, 2 for m and 6 for x.
31
for Example 1
GUIDED PRACTICE
9 b
Solve for b.
STEP 3
Write an equation of the line.
y m x b
Write slope-intercept form.
y 2x 9
Substitute 2 for m and 9 for b.
32
EXAMPLE 2
Write an equation given two points
Write an equation of the line that passes through
( 2, 5) and (2, 1).
SOLUTION
Calculate the slope.
STEP 1
Find the y-intercept. Use the slope and the point
( 2, 5).
STEP 2
y m x b
Write slope-intercept form.
33
EXAMPLE 2
Write an equation given two points
2 b
Solve for b.
Write an equation of the line.
STEP 3
y m x b
Write slope-intercept form.
34
for Examples 2 and 3
GUIDED PRACTICE
2. Write an equation of the line that passes
through (1, 2) and (5, 4).
SOLUTION
Calculate the slope.
STEP 1
Find the y-intercept. Use the slope and the point
(1, 2).
STEP 2
y m x b
Write slope-intercept form.
35
for Examples 2 and 3
GUIDED PRACTICE
Substitute 2 for y and 1 for m.
1 b
Solve for b.
STEP 3
Write an equation of the line.
y m x b
Write slope-intercept form.
y x 1
Substitute 1 for m and 1 for b.
36
EXAMPLE 3
2. Do these three points lie on the same line?
(-4,-2), (2,2.5) and (8,7).
SOLUTION
Calculate the slope.
STEP 1
Find the y-intercept. Use the slope and the point
(-4, 2).
STEP 2
y m x b
Write slope-intercept form.
37
for Examples 2 and 3
GUIDED PRACTICE
Substitute 2 for y and 4 for x and ¾ for m..
1 b
Solve for b.
STEP 3
Write an equation of the line.
y m x b
Write slope-intercept form.
y 3/4 x 1
Substitute 1 for m and 1 for b.
STEP 4
Plug in (8,7) to see if its a solution
(7) ¾ (8) 1 7 6 1 7 7 ? so all three
points are on the line
38

EXAMPLE 4
Solve a multi-step problem
GYM MEMBERSHIP
Your gym membership costs 33 per month after an
initial membership fee. You paid a total of 228
after 6 months Write an equation that gives the
total cost as a function of the length of your
gym membership (in months). Find the total cost
after 9 months.
SOLUTION
STEP 1
Identify the rate of change and starting value.
39

EXAMPLE 4
Solve a multi-step problem
Rate of change, m monthly cost, 33 per month
Starting value, b initial membership fee
STEP 2
Write a verbal model. Then write an equation.
40

EXAMPLE 4
Solve a multi-step problem
STEP 3
Find the starting value. Membership for 6 months
costs 228, so you can substitute 6 for t and 228
for C in the equation C 33t b.
228 33(6) b
Substitute 6 for t and 228 for C.
30 b
Solve for b.
STEP 4
Write an equation. Use the function from Step 2.
41

EXAMPLE 4
Solve a multi-step problem
STEP 4
Write an equation. Use the function from Step 2.
C 33t 30
Substitute 30 for b.
STEP 5
Evaluate the function when t 9.
C 33(9) 30 327
Substitute 9 for t. Simplify.
42
EXAMPLE 5
Solve a multi-step problem
43
EXAMPLE 5
Solve a multi-step problem
SOLUTION
STEP 1
Identify the rate of change and starting value.
Rate of change, m entry fee per race
Starting value, b track membership cost
STEP 2
Write a verbal model. Then write an equation.
44
EXAMPLE 5
Solve a multi-step problem
STEP 3
Calculate the rate of change. This is the entry
fee per race. Use the slope formula. Racer 1 is
represented by (5, 125). Racer 2 is represented
by (8, 170).
STEP 4
Find the track membership cost b. Use the data
pair (5, 125) for racer 1 and the entry fee per
race from Step 3.
45
EXAMPLE 5
Solve a multi-step problem
C mr b
Write the equation from Step 2.
125 15(5) b
Substitute 15 for m, 5 for r, and 125 for C.
50 b
Solve for b.
46
Warm-Up 5.3
47
Lesson 5.3, For use with pages 302-308
Write an equation of the line.
1. passes through (3, 4), m 3
ANSWER
y 3x 5
2. passes through (2, 2) and (1, 8)
ANSWER
y 2x 6
48
Lesson 5.3, For use with pages 302-308
4. Solve and graph the following
inequality 2x1 - 2 6
x -5 OR x 3
ANSWER
3. Multiply both sides of the slope equation
m (y2-y1) (x2-x1)
by (x2-x1). What do you get?
ANSWER
(y2-y1) m (x2-x1)
49
Vocabulary 5.3 Point slope form

• Point-Slope Form
• Fairly rare way of writing a linear equation that
  includes the slope and coordinates of one point.

50
Notes 5.3 Lin. Eqns in point-slope form

• Studied TWO ways to write Lin. Eqns
• Standard Form Ax By C
• Slope-Intercept Form Y Mx b
• The third way is the Point-Slope Form
• Looks like (y-y1) m(x-x1)
• Need two things
• Slope
• ONE point
• Can be converted to Standard Form or
  Slope-Intercept form.
• CAN look different for every point (which is why
  its not used very often!)

51
Examples 5.3

52
EXAMPLE 1
Write an equation in point-slope form
Write an equation in point-slope form of the line
that passes through the point (4, 3) and has a
slope of 2.
Write point-slope form.
y y1 m (x x1)
y 3 2 (x 4)
Substitute 2 for m, 4 for x1, and 3 for y1.
53
EXAMPLE 1
Write an equation in point-slope form
for Example 1
GUIDED PRACTICE
y y1 m (x x1)
Write point-slope form.
y 4 2 (x 1)
Substitute 2 for m, 4 for y, and 1 for x.
54
EXAMPLE 2
Graph an equation in point-slope form
Graph the equation.
55
EXAMPLE 2
Graph an equation in point-slope form
Plot the point (3, 2). Find a second point on
the line using the slope. Draw a line through
both points.
56
EXAMPLE 2
Graph an equation in point-slope form
for Example 2
GUIDED PRACTICE
SOLUTION
Because the equation is in point-slope form, you
know that the line has a slope of 1 and passes
through the point (2, 1).
Plot the point (2, 1). Find a second point on the
line using the slope. Draw a line through both
points.
57
EXAMPLE 3
Use point-slope form to write an equation
Write an equation in point-slope form of the line
shown.
58
EXAMPLE 3
Use point-slope form to write an equation
SOLUTION
STEP 1
Find the slope of the line.
59
EXAMPLE 3
Use point-slope form to write an equation
STEP 2
Write the equation in point-slope form. You can
us either given point.
Method 1
Method 2
Use ( 1, 3).
Use (1, 1).
y y1 m(x x1)
y y1 m(x x1)
y 3 (x 1)
y 1 (x 1)
CHECK
Check that the equations are equivalent by
writing them in slope-intercept form.
y 3 x 1
y 1 x 1
y x 2
y x 2
60
EXAMPLE 3
Use point-slope form to write an equation
for Example 3
GUIDED PRACTICE
STEP 1
Find the slope of the line.
61
EXAMPLE 3
Use point-slope form to write an equation
for Example 3
GUIDED PRACTICE
STEP 2
Write the equation in point-slope form. You can
us either given point.
Method 1
Method 2
Use (2, 3)
Use (4, 4)
y y1 m(x x1)
y y1 m(x x1)
62
Warm-Up 5.4
63
Lesson 5.4, For use with pages 311-316
Write an equation in point-slope form of the line
that passes through the given points.
1. (1, 4), (6, 1)
y 4 (x 1) or y 1 (x 6)
ANSWER
ANSWER
y 2 3(x 1) or y 7 3(x 2)
64
Lesson 5.4, For use with pages 311-316
3. Convert the equation y 5 3(x 3) to
slope-intercept form AND Standard Form
Slope Intercept y 3x 4 Standard Form -3x y
-4
ANSWER

1. Write the following equation on your whiteboard
   2x 3y 6 and graph it on your calculator.
   HINT What form does the equation have to be in
   for the calculator?
2. Multiply both sides of the ORIGINAL equation by 2
   and write that equation down on your whiteboard.
   
3. Now graph the new equation on your calculator.
4. What do you notice?

New equation 4x 6y 12 Graphs are identical!
ANSWER
65
Vocabulary 5.4

• Standard Form of a Linear Equation
• Ax By C

66
Notes 5.4 Write Lin. Eqns in Standard Form

• Standard form is GREAT for graphing intercepts
• Y intercept ? set x 0
• X Intercept ? set y 0
• Can convert point-slope form and slope intercept
  for to standard for by getting all the variables
  on one side and the constants on the other side.
• Equations that have a common multiple or factor
  are still equivalent.

67
Examples 5.4

68
EXAMPLE 1
Write equivalent equations in standard form
Write two equations in standard form that are
equivalent to 2x 6y 4.
SOLUTION
To write another equivalent equation, multiply
each side by 0.5.
To write one equivalent equation, multiply each
side by 2.
4x 12y 8
x 3y 2
69
Write an equation from a graph
EXAMPLE 2
Write an equation in standard form of the line
shown.
SOLUTION
STEP 1
Calculate the slope.
STEP 2
Write an equation in point-slope form. Use (1, 1).
y y1 m(x x1)
Write point-slope form.
y 1 3(x 1)
Substitute 1 for y1, 23 for m and 1 for x1.
70
Write an equation from a graph
EXAMPLE 2
STEP 3
Rewrite the equation in standard form.
3x y 4
Simplify. Collect variable terms on one side,
constants on the other.
71
EXAMPLE 1
for Examples 1 and 2
GUIDED PRACTICE
SOLUTION
To write one equivalent equation,multiply each
side by 2.
2x 2y 6
3x 3y 9
72
Complete an equation in standard form
EXAMPLE 4
EXAMPLE 3
EXAMPLE 4
SOLUTION
STEP 1
Find the value of A. Substitute the coordinates
of the given point for x and y in the equation.
Solve for A.
Ax 3y 2
Write equation.
A(1) 3(0) 2
Substitute 1 for x and 0 for y.
Simplify.
A 2
A 2
Divide by 1.
73
Complete an equation in standard form
EXAMPLE 4
STEP 2
Complete the equation.
2x 3y 2
Substitute 2 for A.
74
Complete an equation in standard form
EXAMPLE 4
EXAMPLE 3
for Examples 3 and 4
Write an equation of a line
GUIDED PRACTICE
Find the missing coefficient in the equation of
the line that passes through the given point.
Write the completed equation.
5. 4xBy 7, (1,1)
SOLUTION
STEP 1
Find the value of B. Substitute the coordinates
of the given point for x and y in the equation.
Solve for B.
4x By 7
Write equation.
4(1) B(1) 7
Substitute 1 for x and 1 for y.
Simplify.
B 3
75
Complete an equation in standard form
EXAMPLE 4
for Examples 3 and 4
GUIDED PRACTICE
STEP 2
Complete the equation.
4x 3y 7
Substitute 3 for B.
76
Complete an equation in standard form
EXAMPLE 4
EXAMPLE 3
for Examples 3 and 4
Write an equation of a line
GUIDED PRACTICE
Find the missing coefficient in the equation of
the line that passes through the given point.
Write the completed equation.
6. Axy 3, (2, 11)
SOLUTION
STEP 1
Find the value of A. Substitute the coordinates
of the given point for x and y in the equation.
Solve for A.
Ax y 3
Write equation.
A(2) 11 3
Substitute 2 for x and 11 for y.
Simplify.
2A 14
Divide each side by 2.
A 7
77
Complete an equation in standard form
EXAMPLE 4
for Examples 3 and 4
GUIDED PRACTICE
STEP 2
Complete the equation.
7x y 3
Substitute 7 for A.
78
Solve a multi-step problem
EXAMPLE 5
Library
Your class is taking a trip to the public
library. You can travel in small and large vans.
A small van holds 8 people and a large van holds
12 people. Your class could fill 15 small vans
and 2 large vans.
b. Graph the equation from part (a).
c. List several possible combinations.
79
Solve a multi-step problem
EXAMPLE 5
SOLUTION
a. Write a verbal model. Then write an equation.
Because your class could fill 15 small vans and 2
large vans, use (15, 2) as the s- and l-values to
substitute in the equation 8s 12l p to find
the value of p.
Substitute 15 for s and 2 for l.
8(15) 12(2) p
144 p
Simplify.
Substitute 144 for p in the equation 8s 12l p.
80
Solve a multi-step problem
EXAMPLE 5
b. Find the intercepts of the graph.
Substitute 0 for s.
8(0) 12l 144
l 12
8s 12(0) 144
Substitute 0 for l.
8s 12(0) 144
s 18
81
Solve a multi-step problem
EXAMPLE 5
Plot the points (0, 12) and (18, 0). Connect them
with a line segment. For this problem only
nonnegative whole-number values of s and l make
sense.
8s 12(0) 144
82
Solve a multi-step problem
EXAMPLE 5
for Example 5
GUIDED PRACTICE
Solve a multi-step problem
EXAMPLE 5
7. WHAT IF? In Example 5, suppose that 8
students decide not to go on the class trip.
Write an equation that models the possible
combinations of small and large vans that your
class could fill. List several possible
combinations.
83
Solve a multi-step problem
EXAMPLE 5
for Example 5
GUIDED PRACTICE
Solve a multi-step problem
EXAMPLE 5
SOLUTION
Write a verbal model. Then write an equation.
STEP 1
8 students decide not to go on the class trip, so
the class could fill 14 small vans and 2 large
vans. Because your class could fill 14 small vans
and 2 large vans, use (14, 2) as the s- and
l-values to substitute in the equation 8s 12l
p to find the value of p.
8(14) 12(2) p
Substitute 14 for s and 2 for l.
136 p
Simplify.
Substitute 136 for p in the equation 8s 12l p.
84
Solve a multi-step problem
EXAMPLE 5
for Example 5
GUIDED PRACTICE
ANSWER
The equation 8s 12l 136 models the possible
combinations.
Find the intercepts of the graph.
STEP 2
Substitute 0 for s.
8(0) 12l 136
8s 12(0) 144
Substitute 0 for l.
8s 12(0) 136
s 17
85
for Example 5
GUIDED PRACTICE
STEP 3
The graph passes through (17, 0), (14, 2), (11,
4), (8, 6), (5, 8) and (2, 10). So, several
combinations are 17 small, 0 large 14 small 2
large 11 small, 4 large 18 small, 6 large 5
small, 8 large 2 small, 10 large.
8s 12(0) 144
86
Warm-Up 5.5
87
Lesson 5.5, For use with pages 318-324
Are the lines parallel? Explain.
1. y 2 2x, 2x y 7
No one slope is 2 and the other is 2.
ANSWER
2. x y 4, 3x 3y 5
ANSWER
Yes both slopes are 1.
2. Graph x y gt 1
ANSWER
88
Lesson 5.5, For use with pages 318-324
3. You play tennis at two clubs. The total cost C
(in dollars) to play for time t (in hours) and
rent equipment is given by C 15t 23 at one
club and C 15t 17 at the other. What is the
difference in total cost after 4 hours of play?
ANSWER
6
3. You play tennis at a club. The first two hours
are free for members if you pay the 50 yearly
fee. After that you pay 10 per hour. Your
friend isnt a member, so she can play for 15 an
hour. Write two equations for the cost as a
function of time played.
Members C(t) 10(t-2) 50 Non-Members C(t)
15t
ANSWER
89
Lesson 5.5, For use with pages 318-324
6. Find Equation in SLOPE INTERCEPT FORM of the
line that goes through (2,8) and (5,17)
Y 3x 2
ANSWER
2. Write equation in POINT-SLOPE FORM of the
line that goes through (2,-6) and (-3, 4)
ANSWER
Y6 -2(x-2) OR y-4 -2(x3)
2. Write an equation in STANDARD FORM of the
line that passes through (1,3) and (3,13)
-5X Y -2
ANSWER
90
5.5 - Warmup

1. Take out your graph paper, patty paper, ruler,
   protractor, and pencil.
2. Graph two points that are on gridlines and
   connect them. MAKE SURE YOUR SLOPE DOES NOT
   EQUAL 1.
3. Find the slope of your line.
4. Take one of the two points, and using the
   protractor make a point that is 90 degrees from
   your point. Connect these points.
5. Find the slope of your new line.
6. What do we call these types of lines?
7. What do you notice about the slopes of
   perpendicular lines?
8. But I bet it only works once!!!! grin

91
Vocabulary 5.5

• Conditional Statement
• A statement with a hypothesis and a conclusion
• Frequently posed as an if-then statement
• Example If Haley is at a volleyball game, then
  shes not here.
• Is this statement always true?
• Converse
• A statement that swaps the hypothesis and
  conclusion of a conditional statement.
• Example If Haleys not here, then shes at a
  volleyball game.
• Is this statement always true?
• Perpendicular lines
• Lines that form a right angle at their
  intersection.

92
Notes 5.5 Parallel and Perpendicular Lines

• If two lines are Parallel, then their slopes
  are???
• Equal
• Whats the converse?? And is it always true??
• If two lines have equal slopes, then they are ???
• Parallel OR
• IDENTICAL!!
• If two lines are Perpendicular, then their slopes
  are ?
• Negative Reciprocals
• Whats the converse?? Is it always true??
• If two non-vertical lines have slopes that are
  negative reciprocals, they are ???
• Perpendicular
• Yes, its always true!

93
Examples 5.5

94
Write an equation of a parallel line
EXAMPLE 1
Write an equation of the line that passes through
(3,5) and is parallel to the line y 3x 1.
SOLUTION
STEP 1
Identify the slope. The graph of the given
equation has a slope of 3. So, the parallel line
through ( 3, 5) has a slope of 3.
STEP 2
Find the y-intercept. Use the slope and the given
point.
95
Write an equation of a parallel line
EXAMPLE 1
y mx b
Write slope-intercept form.
5 3( 3) b
Substitute 3 for m, 23 for x, and 25 for y.
Solve for b.
4 b
STEP 3
Write an equation. Use y mx b.
Substitute 3 for m and 4 for b.
y 3x 4
96
for Example 1
GUIDED PRACTICE
SOLUTION
STEP 1
Identify the slope. The graph of the given
equation has a slope of 1.So, the parallel line
through ( 2, 11) has a slope of 1.
STEP 2
Find the y-intercept. Use the slope and the given
point.
97
for Example 1
GUIDED PRACTICE
y mx b
Write slope-intercept form.
11 (1 )( 2) b
Substitute 11 for y, 1 for m, and 2 for x.
Solve for b.
9 b
STEP 3
Write an equation. Use y m x b.
y x 9
Substitute 1 for m and 9 for b.
98
Determine whether lines are parallel or
perpendicular
EXAMPLE 2
Determine which lines, if any, are parallel or
perpendicular.
Line a y 5x 3
Line b x 5y 2
Line c 10y 2x 0
SOLUTION
Find the slopes of the lines.
Line a The equation is in slope-intercept form.
The slope is 5. Write the equations for lines b
and c in slope-intercept form.
99
Determine whether lines are parallel or
perpendicular
EXAMPLE 2
Line b
x 5y 2
5y x 2
Line c
10y 2x 0
10y 2x
100
Determine whether lines are parallel or
perpendicular
EXAMPLE 2
101
for Example 2
GUIDED PRACTICE
Determine which lines, if any, are parallel or
perpendicular.
Line a 2x 6y 3
Line b 3x 8 y
Line c 1.5y 4.5x 6
Find the slopes of the lines.
Line a 2x 6y 3
6y 2x 3
102
for Example 2
GUIDED PRACTICE
Line b 3x 8 y
Line c 1.5y 4.5x 6
1.5y 4.5x 6
y 3x 4
103
Determine whether lines are perpendicular
EXAMPLE 3
Line a 12y 7x 42
Line b 11y 16x 52
SOLUTION
Find the slopes of the lines. Write the equations
in slope-intercept form.
104
Determine whether lines are perpendicular
EXAMPLE 3
Line a 12y 7x 42
Line b 11y 16x 52
105
EXAMPLE 4
Write an equation of a perpendicular line
Write an equation of the line that passes through
(4, 5) and is perpendicular to the line y 2x
3.
SOLUTION
STEP 1
106
Write an equation of a perpendicular line
EXAMPLE 4
STEP 2
Find the y-intercept. Use the slope and the given
point.
Write slope-intercept form.
Solve for b.
STEP 3
Write an equation.
y m x b
Write slope-intercept form.
107
for Examples 3 and 4
GUIDED PRACTICE
3. Is line a perpendicular to line b? Justify
your answer using slopes
Line a 2y x 12
Line b 2y 3x 8
SOLUTION
Find the slopes of the lines. Write the equations
in slope-intercept form.
Line a 2y x 12
108
for Examples 3 and 4
GUIDED PRACTICE
Line b 2y 3x 8
109
for Examples 3 and 4
GUIDED PRACTICE
4. Write an equation of the line that passes
through (4, 3) and is perpendicular to the
line y 4x 7.
SOLUTION
STEP 1
110
for Examples 3 and 4
GUIDED PRACTICE
Find the y-intercept. Use the slope and the
given point.
STEP 2
Write slope-intercept form.
Solve for b.
Write an equation.
STEP 3
y m x b
Write slope-intercept form.
111
Warm-Up 5.6
112
Lesson 5.6, For use with pages 325-333
Find the slopes of the line that passes through
the point.
1. (4, 1) and (6, 4)
2. (2, 3) and (1, 6)
ANSWER
3
113
Lesson 5.6, For use with pages 325-333
Find the slopes of the line that passes through
the point.
3. Your commission c varies with the number s of
pair of shoes you sell. You made 180 when you
sold 15 pairs of shoes. Write a direct variation
equation that relates c to s.
ANSWER
c 12s
3. Graph 3x 4y lt 12
ANSWER
114
Vocabulary 5.6

• Scatter Plot
• Points that show relationships or trends in data
• Correlation between data
• Shows a relationship between data (if it exists)
• Line of fit
• AKA Linear Regression
• Line that approximates the data by modeling the
  trend.

115
Notes 5.6 Fit Data to a Line

• Three types of correlation
• Positive Trends Up As x gets larger, y gets
  larger
• Negative Trends Down As x gets larger, y gets
  smaller
• No Trend Little to no pattern or correlation

Negative
No Trend
Positive
116
Notes 5.6 Fit Data to a Line

• To construct a regression line, about half the
  points should be above the line and about half
  below.
• The accuracy of the line or how close is models
  the data is given by the r2 value (r is short for
  residuals.)
• If the r2 value is VERY close to 1 the regression
  line accurately models the data.
• If the r2 value is not close, the data isnt
  showing a very strong correlation and the
  regression line is not very accurate.
• Once you have the line, you can use points or the
  graph to determine the equation of the line.

117
Examples 5.6

118
EXAMPLE 1
Describe the correlation of data
Describe the correlation of the data graphed in
the scatter plot.
119
EXAMPLE 1
Describe the correlation of data
b. The scatter plot shows a negative
correlation between hours of television watched
and test scores. that as the hours of television
This means that as the hours of television
watched Increased, the test scores tended to
decrease.
120
for Example 1
GUIDED PRACTICE
121
EXAMPLE 2
Make a scatter plot
Swimming Speeds
The table shows the lengths (in centimeters) and
swimming speeds (in centimeters per second) of
six fish.
122
EXAMPLE 2
Make a scatter plot
a. Make a scatter plot of the data.
b. Describe the correlation of the data.
123
EXAMPLE 2
Make a scatter plot
SOLUTION
124
for Example 2
GUIDED PRACTICE
125
EXAMPLE 3
Write an equation to model data
BIRD POPULATIONS
The table shows the number of active red-cockaded
woodpecker clusters in a part of the De Soto
National Forest in Mississippi. Write an equation
that models the number of active clusters as a
function of the number of years since 1990.
Year 1992 1993 1994 1995 1996 1997 1998 1999 2000
Active clusters 22 24 27 27 34 40 42 45 51
126
EXAMPLE 3
Write an equation to model data
SOLUTION
STEP 1
127
EXAMPLE 3
Write an equation to model data
STEP 2
Decide whether the data can be modeled by a line.
Because the scatter plot shows a positive
correlation, you can fit a line to the data.
STEP 3
Draw a line that appears to fit the points in
the scatter plot closely.
STEP 4
Write an equation using two points on the line.
Use (2, 20) and (8, 42).
128
EXAMPLE 3
Write an equation to model data
Find the slope of the line.
Find the y-intercept of the line. Use the point
(2, 20).
Write slope-intercept form.
129
EXAMPLE 3
Write an equation to model data
Solve for b.
An equation of the line of fit is y
130
for Example 3
GUIDED PRACTICE
131
EXAMPLE 4
Interpret a model
Refer to the model for the number of woodpecker
clusters in Example 3.
a. Describe the domain and range of the
function.
b. At about what rate did the number of active
woodpecker clusters change during the period
19922000?
132
EXAMPLE 4
Interpret a model
SOLUTION
133
EXAMPLE 4
for Example 4
GUIDED PRACTICE
134
Warm-Up 5.7
135
Lesson 5.7, For use with pages 334-342
1. Evaluate f(x) 2.5x 8 when x is 3 or 5.
15.5 20.5
ANSWER
2. The table shows the profit of a company. Write
an modeling the profit y as the function
of the number of years x since 1998.
ANSWER
y 2.8x 16.4
136
Vocabulary 5.7

• Linear Regression
• Line of best fit Approximates the data in
  plot
• Interpolation
• Using a linear regression line to APPROXIMATE a
  point that is BETWEEN TWO KNOWN VALUES!
• Extrapolation
• Using a linear regression line to APPROXIMATE a
  point that is OUTSIDE of the data range (OR THE
  KNOWN VALUES).
• Tells the future!!
• Zero of a function
• Where a FUNCTION equals zero.

137
Notes 5.7 Predict with Lin. Models

• You can use your knowledge of Linear functions
  and the calculator to predict the future or the
  past.
• IMPORTANT CALCULATOR FUNCTIONS
• Enter Lists STAT?EDIT
• Plot Lists
• 2nd Y (Stat Plot)
• Turn Plot on and configure it (SET WINDOW!!)
• Remember ZOOM?STATPLOT as well.
• Determine points on a line (e.g. to find y1(x)
  using function notation or predict the future!)
• VARS?Y-Vars?Function?Y1 Enter
• Press ( put value here ) and press Enter
• Line should look like Y1(5)

138
Notes 5.7 Continued

• IMPORTANT CALCULATOR FUNCTIONS
• To find a Regression Line
• Stat?Calc?Linreg (axb)
• Type in the lists where you put the data
• Tell Calli where you want to store the eqn.
• EX LinReg (axb) L1,L2,Y1
• Turn Plot on and configure it (SET WINDOW!!)
• Remember ZOOM?STATPLOT as well.

139
Examples 5.7 TURN TO PAGE 335 IN BOOK!

140

EXAMPLE 1
Interpolate using an equation
CD SINGLES
The table shows the total number of CD single
shipped (in millions) by manufacturers for
several years during the period 19931997.
141

EXAMPLE 1
Interpolate using an equation
a.
Make a scatter plot of the data.
142

EXAMPLE 1
Interpolate using an equation
SOLUTION
143

EXAMPLE 1
Interpolate using an equation
144

EXAMPLE 2
Extrapolate using an equation
CD SINGLES
Look back at Example 1.
145

EXAMPLE 2
Extrapolate using an equation
SOLUTION
146

EXAMPLE 2
Extrapolate using an equation
147
for Examples 1 and 2
GUIDED PRACTICE
a. Find an equation that models the floor area
(in square feet) of a new single-family house as
a function of the number of years since 1995.
148
for Examples 1 and 2
GUIDED PRACTICE
b. Predict the median floor area of a new
single-family house in 2000 and in 2001.
c. Which of the predictions from part (b) would
you expect to be more accurate? Explain your
reasoning.
149
EXAMPLE 3
Predict using an equation
SOFTBALL
The table shows the number of participants in
U.S. youth softball during the period 19972001.
Predict the year in which the number of youth
softball participants reaches 1.2 million.
Year 1997 1998 1999 2000 2001
Participants (millions) 1.44 1.4 1.411 1.37 1.355
150
EXAMPLE 3
Predict using an equation
SOLUTION
STEP 1
Perform linear regression. Let x represent the
number of years since 1997, and let y represent
the number of youth softball participants (in
millions). The equation for the best-fitting line
is approximately y 0.02x 1.435.
151
EXAMPLE 3
Predict using an equation
STEP 2
Graph the equation of the best-fitting line.
Trace the line until the cursor reaches y 1.2.
The corresponding x-value is shown at the bottom
of the calculator screen.
152
for Example 3
GUIDED PRACTICE
2. SOFTBALL In Example 3, in what year will there
be 1.25 million youth softball participants in
the U.S?
153
EXAMPLE 4
Find the zero of a function
SOFTBALL
Look back at Example 3. Find the zero of the
function. Explain what the zero means in this
situation.
SOLUTION
Substitute 0 for y in the equation of the
best-fitting line and solve for x.
y 0.02x 1.435
Write the equation.
0 0.02x 1.435
Substitute 0 for y.
Solve for x.
154
EXAMPLE 4
Find the zero of a function
155
for Example 4
GUIDED PRACTICE
SOLUTION
Substitute 0 for y in the equation of the
best-fitting line and solve for x.
y 1.23x 14
Write the equation.
0 1.23x 14
Substitute 0 for y.
Solve for x.
156
EXAMPLE 4
for Example 4
GUIDED PRACTICE
157
Review Ch. 5
158

Daily Homework Quiz
For use after Lesson 5.1
Write an equation of the line that passes through
the given points.
159

Daily Homework Quiz
For use after Lesson 5.1
160

Daily Homework Quiz
For use after Lesson 5.2
Write an equation of the line that passes through
the given point with given slope.
161

Daily Homework Quiz
For use after Lesson 5.2
Write an equation of the line that passes through
the given point.
162

Daily Homework Quiz
For use after Lesson 5.2
163

Daily Homework Quiz
For use after Lesson 5.2
164
Daily Homework Quiz
For use after Lesson 5.4
Write an equation in standard form of the line
that passes through the given point and has the
given slope m or that passes through the two
given points.
2. ( 4, 3), (2, 9)
165
Daily Homework Quiz
For use after Lesson 5.4
166
Daily Homework Quiz
For use after Lesson 5.5
167
Daily Homework Quiz
For use after Lesson 5.5
168
Daily Homework Quiz
For use after Lesson 5.6
1. Tell whether x and y show a positive
correlation, a negative correlation, or
relatively no correlation.
169
Daily Homework Quiz
For use after Lesson 5.6
170
Warm-Up X.X
171
Vocabulary X.X

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172
Notes X.X LESSON TITLE.

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173
Examples X.X