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Algebra
Chapter 5: LINEAR FUNCTIONS
Name:______________________________
Teacher:____________________________
Pd: _______
Table of Contents
Day 1 - Chapter 5-3/5-4: Slope SWBAT: Calculate the slope from any two points
Pgs. #1 - 5
Hw pgs. #6 – 7
Day 2 - Chapter 5-6: Slope – Intercept Form SWBAT: Write and Graph a linear equation in Slope – Intercept Form
Pgs. #8 – 11
Hw pgs. #12 – 14
Day 3 - Chapter 5-7: Point - Slope Form SWBAT: Write and Graph a linear equation in Point - Slope Form
Pgs. #15 – 19
Hw pgs. #20 – 21
Day 4 - Chapter 5-6/5-7: Vertical and Horizontal Lines
SWBAT: Write and Graph Vertical and Horizontal Lines
Pgs. #22-25
Hw pgs. #26-27
Day 5 - Review: Sections 5-2 through 5 - 7 Pgs. #28-31
Day 6 - Chapter 5-8: Slopes of Parallel and Perpendicular Lines SWBAT: Calculate the slope of Parallel and Perpendicular Lines
Pgs. #32-37
Hw pgs. #38-39
Day 7 - Chapter 5-8: Equations of Parallel and Perpendicular Lines SWBAT: Write the equation of Parallel and Perpendicular Lines
Pgs. #40-43
Hw pgs. #44-45
Day 8 - Chapter 5-5: Direct Variation SWBAT: Identify, write, and graph direct variations
Pgs. #46-50
Hw pgs. #50-51
Day 9 – 10: Review of Chapter 5 SWBAT: Graph and Write Linear Equations
Pgs. #52-66
1
Day 1 – Slope
Warm - Up
Find the x- and y- intercepts.
a. b.
x-intercept: _______________ x-intercept: _______________
y-intercept: _______________ y-intercept: _______________
Motivation:
The slope of a line measures the steepness of the line.
We’re familiar with the word slope as it relates to mountains.
Skiers and snowboarders refer to “hitting the slopes.”
Slope measures the ratio of the change in the y-value of a line to a
given change in its x-value.
Finding Slope
DEFINITION OF SLOPE
2
Example 1: Calculating the slope from a Graph
a: Find the slope of the line. b: Find the slope of the line.
1st Point: _________ 2
nd Point: ________ 1
st Point: _________ 2
nd Point: _________
2 1
2 1
slopeY Y
X X
ym
x
2 1
2 1
slopeY Y
X X
ym
x
c: Find the slope of the line. d: Find the slope of the line.
1st Point: _________ 2
nd Point: _________ 1
st Point: ________ 2
nd Point: _________
2 1
2 1
slopeY Y
X X
ym
x
2 1
2 1
slopeY Y
X X
ym
x
x
y
x
y
x
y
x
y
3
e. f.
Example 2: Calculating the slope from a set up points
a) Find the slope of the line that passes through the points 8, 7 and 4, 5 .
2 1
2 1
Y Y
X Xm
and
4
and
b) Find the slope of the line that passes through the points (4, 3) and (–5, –2).
2 1
2 1
Y Y
X Xm
c) Find the slope of the line that passes through the points (–4, –7) and (–3, –5).
2 1
2 1
Y Y
X Xm
Example 3: Calculating the Slope from a Table of Values
a) Find the slope of the line that contains the points from the table
b) Find the slope of the line that contains the points from the table
and
5
Challenge
Summary
Exit Ticket
1. Fill in the blanks.
2.
6
Day 1 – Homework
Slope
1) Find the slope of the line that passes through each of the following sets of points.
(a) (b) (c)
(d) (e) (f)
2) Find the slope of each of the following lines graphically:
a) fkdjflkdjfjkd b)
7
c) d)
8
Day 2 - Slope – Intercept Form
Warm Up
Motivation:
Who uses this?
9
Graphing by using slope and y-intercept
1. 24
3 xy m = ______ 2. 42 xy m = ______
y-intercept = b = (0, ____ ) y-intercept = b = (0, ____ )
Practice
Graphing by using slope and y-intercept
3. 32
1 xy m = ______ 4. xy 3 m = ______
y-intercept = b = (0, ____ ) y-intercept = b = (0, ____ )
10
5.
Writing Equations of Graphs
9. Write the equation in slope–intercept form. Then graph the line described by the equation.
3x + 2y = 5
6. 7. 8.
11
10. Write the equation in slope–intercept form. Then graph the line described by the equation.
4x - 2y = 14
Challenge
SUMMARY
Exit Ticket
12
Day 2 – Homework
Slope – Intercept Form
1) Which of the following lines has a slope of 5 and a y-intercept of 3 ?
(1) 5 3y x (3) 3 5y x
(2) 5
3y x (4) 3 5y x
2) Which of the following equations represents the graph shown?
(1) 3
32
y x (3) 2
33
y x
(2) 3
22
y x (4) 2
23
y x
3) Graph each line.
(a) y = –3x+ 2 (b) y = 1
2 x
y
x
13
(c) y = 6x + 3 (d) y = 2
3x + 9
(e) y = –x + 2 (f) y = x - 3
(g) y = -
x - 4 (h) y = 2x + 7
14
1) Write an equation of each line in slope – intercept form. Identify the slope and y-intercept.
a. 2 2 4x y b. 5 7x y
c. 6x + 2y = 8 d. 10 5 15y
15
Day 3 – Point Slope Form
Warm – Up
16
Yesterday, we learned how to graph equations using the slope and the y-intercept. Today we are
going to write equations of lines.
First, let’s see how to use the equation.
Example 1: Graph the linear equation.
y – 1 = 2(x – 3) m = _______ pt = ( ___, ____ ) Practice: Graph the linear equation.
y + 4 = -¼(x – 8) m = _______ pt = ( ___, ____ )
Equations of Lines
17
Example 2: Converting Point-Slope to Slope-Intercept Form and Standard Form.
Write in Slope-Intercept form and Standard Form.
Practice: Converting Point-Slope form to Slope intercept Form
Writing Equations of Lines
Example 3: Write the equation of a line given the slope and a point.
(-3, -4); m = –3
18
Practice: Write the equation of a line given the slope and a point.
(1, 2); m = –3
Example 4: Write the equation of a line passing through the two points given.
(10, 20) and (20, 65)
Step 1:
Step 2: plug m, and point into equation.
y – y1 = m(x – x1)
Practice: Write the equation of a line passing through the two points given.
(2, –5) and (–8, 5)
Step 1:
Step 2: plug m, and point into equation.
y – y1 = m(x – x1)
19
Challenge
Write the equation of a line in point-slope form passing through the two points given.
(f, g) (h, j)
SUMMARY
Exit Ticket
20
Day 3 – Point Slope Form - HW
1) Which equation describes the line through (–5, 1) with the slope of 1?
(a) y = x – 6 (c) y = –5x + 6
(b) y = –5x – 6 (d) y = x + 6
2) A line contains (4, 4) and (5, 2). What is the slope and y – intercept?
(a) slope = –2; y – intercept = 2 (c) slope = –2; y – intercept = 12
(b) slope = 1.2; y – intercept = –2 (d) slope = 12; y – intercept = 1.2
Write an equation for the line with the given slope and point in slope-intercept form.
3) slope = 3; (–4, 2)
Equation: ______________________
4) slope = –1; (6, –1 )
Equation: ______________________
5) slope = 0; (1, –8)
Equation: ______________________
6) slope = –9; (–2, –3)
Equation: ______________________
21
Write an equation for the line through the two points in slope intercept form.
7) (2, 1); (0, –7)
Equation: ______________________
8) (–6, –6); (2, –2)
Equation: ______________________
9) (–2, –3); (–1, –4)
Equation: ______________________
10) (6, 12); (0, 0)
Equation: ______________________
Write an equation for the line for each graph.
11) 12)
22
Day 4 – Vertical and Horizontal Lines
Warm – Up:
Horizontal Lines Picture:
Slope =
If your point is (a, b), then your equation is always y = ____
Ex: Write the equation of the horizontal line passing through each point.
1. (3, 7)
2. (2, -4) 3. (8, 0)
4.
5. 6.
23
Vertical Lines Picture:
Slope =
If your point is (a, b), then your equation is always x = ____
Ex: Write the equation of the vertical line passing through each point.
7. (2, 9)
8. (1, -3) 9. (0, 10)
10.
11. 12.
Parallel Lines
Parallel Lines have the _______ slope What can you say about all horizontal lines?
24
What can you say about all vertical lines? What can you say about a horizontal line and a vertical line?
Write the equation of the line…..
13. Parallel to y = 4, through (2, -3)
14. Parallel to x = -3, through (1, 2)
15. Parallel to the y-axis through (2, 4)
16. Parallel to the x-axis through (3, -5)
17. Through (2, 8) with a slope of 0.
18. Through (3, 3) with an undefined slope.
25
Write the slope-intercept form of the equation of the line through the given points.
1) through: (5, -2) and (0, -2) 8) through: (-6, -1) and (-6, 11)
Challenge Write the equation of a line perpendicular to y = 1, passing through (2, 10). SUMMARY Memory Device for Vertical Lines Memory Device for Horizontal Lines
x = “a” value of point = (a, b) y = “b” value of point = (a, b)
Exit Ticket
1.
2.
26
Day 4 - HW
27
13) Write the equation of a line parallel to x = 12 passing through the point (4, 5).
14) Write the equation of a line parallel to y = 1 passing through the point (-3, 7).
15) Write the equation of a line parallel to the x-axis passing through the point (1, 1).
16) Write the equation of a line parallel to the y-axis passing through the point (-8, 0).
28
Day 5 – Review of Sections 5-2 through 5-6
29
30
31
Write the equation of the lines below. Write your answer in slope-intercept form.
32
Day 6 – Slopes of Parallel and Perpendicular Lines
Warm Up
Directions: Find the reciprocal.
1) 2 2) 3
1 3)
9
5
Directions: Find the slope of the line that passes through each pair of points.
4) (2, 2) and (–1, 3) 5) (3, 4) and (4, 6) 6) (5, 1) and (0, 0)
33
34
35
Example 3:
Graph a line parallel to the given line and passing through the given point.
Graph a line perpendicular to the given line and passing through the given point.
36
37
Challenge
If ||PQ RS and the slope of 1
4
xPQ
and the slope of RS is
6
8, then find the value of x. Justify
algebraically or numerically.
SUMMARY
Exit Ticket
1.
2.
38
Day 6 – Slopes of Parallel and Perpendicular Lines HW
39
7)
8)
9)
10)
40
Day 7 – Equations of Parallel and Perpendicular Lines
Warm Up
Determine the equation of the line that passes through the two points 2,4 and 8,0 .
1) Write an equation for the line that passes through (4, 10) and is parallel to the line described by y = 3x + 8.
Step 1: m = _____
Step 2: plug in the point into y – y1 = m(x – x1) and solve for y.
Equation: __________________________________
2) Write an equation for the line that passes through (–2 , 5) and is parallel to the line described by y = 2
1x – 7.
Step 1: m = _____
Step 2: plug in the point into y – y1 = m(x – x1) and solve for y.
Equation: __________________________________
41
3) Write an equation for the line that passes through (2, –1) and is perpendicular to the line described by
y = 2x – 5.
Step 1: m = _____
Step 2: plug in the point into y – y1 = m(x – x1) and solve for y.
Equation: __________________________________
4) Write an equation for the line that passes through (2, 6) and is perpendicular to the line described by
y = -3
1x + 2.
Step 1: m = _____
Step 2: plug in the point into y – y1 = m(x – x1) and solve for y.
Equation: __________________________________
42
Regents Practice
5) What is the slope of a line parallel to the line whose equation is y = -4x + 5?
6) What is the slope of a line parallel to the line whose equation is 3x + 6y = 6?
7)
8) Kjk
43
Challenge
SUMMARY
Exit Ticket
1.
2.
44
Day 7 – Equations of Parallel and Perpendicular Lines - HW
45
46
Day 8 – Direct Variation
Warm Up
Find the slope of the linear function represented by the table below.
1)
2)
Motivation:
Who uses this?
A recipe for paella calls for 1 cup of rice to make 5 servings. In other words, a chef
needs 1 cup of rice for every 5 servings.
The equation y = 5x describes this relationship. In this relationship, the number of
servings varies directly with the number of cups of rice.
47
Direct Variation
Example1: Identifying Direct Variations from an Equation
Tell whether the equation represents a direct variation. If so, identify the constant of
variation.
1) y = 21x 2) 3x + y = 8
Direct Variation? _____________ Direct Variation? ____________
Constant of Variation: ___________ Constant of Variation: __________
3) –4x + 3y = 0 4) 3x + 0 = 15y
Direct Variation? _____________ Direct Variation? ____________
Constant of Variation: ___________ Constant of Variation: __________
What happens if you solve y = kx for k?
Let’s see…
In a direct variation, the ratio ________
is equal to the constant of variation.
48
Example 2: Identifying Direct Variations from Ordered Pairs
Tell whether the relationship is a direct variation. Explain.
5) 6)
Find y
x for each ordered pair. Find
y
x for each ordered pair.
Direct Variation? _____________ Direct Variation? ____________
Constant of Variation: ___________ Constant of Variation: __________
Example 3: Identifying Direct Variations from a Graph
Tell whether the relationship is a direct variation. Explain.
7) 8)
Direct Variation? _____________ Direct Variation? ____________
Constant of Variation: ___________ Constant of Variation: __________
49
Example 4: Writing and Solving Direct Variation Equations
9) The value of y varies directly with x, and y = 3 when x = 9. Find y when x = 21.
Use a proportion: 9
3
x
y
10) The value of y varies directly with x, and y = 4.5 when x = 0.5. Find y when x = 10.
Challenge
SUMMARY
50
Exit Ticket
Day 8 – Homework
Direct Variation
Tell whether each equation is a direct variation. If so, identify the constant of variation.
2) y = 3x 3) y = 2x – 9
3) 2x + 3y = 0 4) 3y = 9x
Find the value of y
xfor each ordered pair. Then tell whether the relationship is a direct variation.
5) 6) 7)
________________ _________________ _________________
51
Tell whether each graph is a direct variation. If so, identify the constant of variation.
8) 9)
_______________ ________________
10) The value y varies directly with x, and y = –18 when x = 6. Find y when x = –8.
11) The value of y varies directly with x and y = ½ when x = 5. Find y when x = 30.
12)
52
Chapter 5: Review of Linear Functions
Section 1: Slope
53
Section 2: X and Y – Intercepts
9. Find the x-intercept and the y-intercept for each of the following:
a. 3x + y = 6 b. -2x – 4y = -8
10. Find the x-intercept and the y-intercept for each of the following:
54
Section 3: Direct Variation
11.
12.
13. Tell whether each is a direct variation. If yes, identify the constant of variation?
55
14.
15.
SECTION 4: Writing Equations of Lines
56
16. 17. 18.
19. 20. 21.
22. 23.
24. 25.
57
26. 27.
28. 29.
58
Graph Each Vertical and/or Horizontal Line.
30. x = 3 31. x = -2
32. y = 4 33. y = -3
34. 35.
59
Applications
36. The graph below shows the relationship between the number of miles driven and the cost of
Super-City’s taxi fare.
Part A
Write an equation show the relationship between the number of miles driven and the cost of
Super-City’s taxi fare.
Equation: __________________
Part B
What is the taxi fare for being driven 15 miles?
Answer: __________________ dollars
Super – City Taxi Fare
1 2 3 4 5 6 7 8 9 10 0
$2
$4
$6
$8
$10
$12
$14
$16 C
ost
(doll
ars)
t
Number of Miles Driven
60
37. The accompanying graph represents the yearly cost of joining a Sports Club.
Part A
Write an equation show the relationship between the number of months and
the yearly cost.
Equation: __________________
Part B
What is the total cost of joining the club and attending 10 months during
the year?
Answer: _______________ dollars
61
38. An online video store rents movies for $2.00 per video plus a $5.00 membership fee.
Part A
Write an equation to represent the cost of renting a video in relationship to the number of videos rented.
Equation: _____________________________
Part B
Make a table of values to represent your equation in part A.
Number of Videos Cost
($)
Part C
Graph your equation using your table of values.
Part D
Use the graph to determine the cost for renting 6 videos.
Answer: __________________ dollars
2
4
6
8
10
12
14
16
18
20
0 1 2 3 4 5 6
Number of videos
Cost for Video Rentals
Cost ($)
62
39. Mrs. Jackson will need a babysitter for her infant baby. She decides to offer $20 for
travel plus $5 per hour worked.
Part A
Write an equation to represent the cost for a babysitter in relationship to the number of hours worked.
Equation: _____________________________
Part B
Make a table of values to represent your equation in part A.
Number of Hours Cost
($)
Part C
Graph your equation using your table of values.
Part D
Determine the cost of hiring a babysitter for 10 hours.
Answer: __________________ dollars
Number of Hours
$ C
ost
63
40. m = -4, (-2, -8)
41.
42.
43.
44.
45.
64
Section 5: Parallel and Perpendicular Lines
46. 47.
48. 49.
50. 51.
65
52. 53.
54. 55. Parallel to x = -2, through (-12, 0)
56. 57. Perpendicular to y = -5, through (1, -2)
58. 59. Parallel to x-axis, through, (3, 8)
(-6, 8)
66
Draw a line parallel to the given line below passing through given point.
60. 61.
Draw a line perpendicular to the given line below passing through given point.
62. 63.
Line1
Line1
Line1
Line1
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