Name Date Ms - White Plains Middle School · 1 Chapter 5 - 2 Day 1 - Using Intercepts to Graph...

Algebra

Chapter 5: LINEAR FUNCTIONS

Name:______________________________

Teacher:____________________________

Pd: _______

Table of Contents

Chapter 5-2: x- and y- Intercepts SWBAT: Use intercepts to graph lines

Pgs. #1 - 4

Hw pgs. #5 – 6

Chapter 5-3/5-4: Slope SWBAT: Calculate the slope from any two points

Pgs. #7 - 11

Hw pgs. #12 – 13

Chapter 5-5: Direct Variation SWBAT: Identify, write, and graph direct variations

Pgs. #14 - 18

Hw pgs. #18 – 19

Chapter 5-6: Slope – Intercept Form SWBAT: Write and Graph a linear equation in Slope – Intercept Form

Pgs. #20 – 24

Hw pgs. #25 – 27

Chapter 5-7: Point - Slope Form SWBAT: Write and Graph a linear equation in Point - Slope Form

Pgs. #28 – 32

Hw pgs. #33 – 34

Review: Sections 5-2 through 5 - 7

Pgs. #35 – 38

Chapter 5-8: Slope of Parallel and Perpendicular Lines SWBAT: Calculate the slope of Parallel and Perpendicular Lines

Pgs. #39 – 42

Hw pgs. #43 – 44

Chapter 5-8: Equations of Parallel and Perpendicular Lines SWBAT: Write the equation of Parallel and Perpendicular Lines

Pgs. #45 – 48

Hw pgs. #49 – 50

1

Chapter 5 - 2

Day 1 - Using Intercepts to Graph Lines

Warm Up

Solve each equation for the variable.

1) 5x + 0 = –10 2) 33 = 0 + 3y

Example 1: Find the x-intercept and y-intercept from the graph.

1. 2.

x-intercept: _____________ x-intercept: _____________

y-intercept: _____________ y-intercept: _____________

Definitions:

The y-intercept is the y-coordinate of the point where the graph intersects the y-axis.

The x-intercept is the x-coordinate of the point where the graph intersects the x-axis.

y

x x

y

2

Practice: Find the x-intercept and y-intercept from the graph.

3. gf 4.

x-intercept: _____________ x-intercept: _____________

y-intercept: _____________ y-intercept: _____________

Example 2: Find the x-intercept and y-intercept from the equation and then graph.

5. 5x – 2y = 10

x- intercept

To find the x-intercept:

replace y with 0 and solve for x.

y-intercept

To find the y-intercept:

replace x with 0 and solve for y

5x – 2y = 10

The x-intercept is __________

5x – 2y = 10

The y-intercept is __________

6. –10x + 6y = 30

3

Practice: Find the x-intercept and y-intercept from the equation and then graph.

7. 4x + 2y = 8

8. 3x – 7y = 21

9.

4

Challenge

SUMMARY

Exit Ticket

5

Day1 – Homework

Using Intercepts to Graph Lines

Find the x-intercept and y-intercept for the graphs below.

1) 2)

x-intercept: _______________ x-intercept: _______________

y-intercept: _______________ y-intercept: _______________

3) 4)



6

Determine the x-intercept and y-intercept for each equation and graph.

5) 3x + 2y = –6 6) x – 4y = 4

7) y = 0.25x – 1 8) 2x + 3y = 6

7

Chapter 5 – 3/5 - 4

Day 2 – Slope

Warm - Up

Find the x- and y- intercepts.

a. b.



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 1

2 1

rise change in slope

run change in

Y Y

X X

y ym

x x

8

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

9

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

1,1

)7,8(yx

and

2,2

)5,4(yx

10

1,1

)3,4(yx

and

2,2

)2,5(

yx

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

1,1

)7,4(

yx

and 2,2

)5,3(

yx

11

How to remember slope?

Ski - Slope House

Challenge

Summary

Exit Ticket

1. Fill in the blanks.

2.

12

Day2 – 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)

13

c) d)

14

Chapter 5 - 5

Day 3 – 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.

15

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



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.

16

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.



Example 3: Identifying Direct Variations from a Graph

Tell whether the relationship is a direct variation. Explain.

7) 8)



17

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

18

Exit Ticket

Day 3 – Homework

Direct Variation

Tell whether each equation is a direct variation. If so, identify the constant of variation.

1) y = 3x 3) y = 2x – 9

2) 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)

________________ _________________ _________________

19

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)

20

Chapter 5 – 6

Day 4 - Slope – Intercept Form

Warm Up

Motivation:

Who uses this?

21

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, ____ )

22

5. y = –4

Make a Table of Values:

x y

What do you notice about the y values? __________

Graph and label the lines 5 and 2y y on the same grid.

6. x = 2

Make a Table of Values:

x y

What do you notice about the x values? __________

Graph and label the lines x = –1 and x = 5 on the same grid.

EQUATIONS OF HORIZONTAL LINES

where (or simply ) 0 y by mx b m

y

x

EQUATIONS OF VERTICAL LINES

where is the of the lineintercepta xx a

23

7.

Writing Equations of Graphs

11. Write the equation in slope–intercept form. Then graph the line described by the equation.

3x + 2y = 5

8. 9. 10.

24

12. Write the equation in slope–intercept form. Then graph the line described by the equation.

4x - 2y = 14

Challenge

SUMMARY

Exit Ticket

25

Day 4 – 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+ 0

y

x

26

(c) y = 6x + 3 (d) y = 2

3x + 9

(e) y = –1 (f) x = –5

(g) y = 8 (h) x = 7

27

3) 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

28

Day 5 – Point Slope Form

Warm – Up

29

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

30

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

31

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)

32

Challenge

Write the equation of a line in point-slope form passing through the two points given.

(f, g) (h, j)

SUMMARY

Exit Ticket

33

Day 5 – 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: ______________________

34

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)

35

Day 6 – Review of Sections 5-2 through 5-6

38

Write the equation of the lines below. Write your answer in slope-intercept form.

39

Day 7 – Slopes of Parallel and Perpendicular Lines (5 – 8)

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)

41

m m

m =

m =

m =

m =

m =

m =

m =

m =

m =

m =

m =

m =

42

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.

43

Day 7 – Slopes of Parallel and Perpendicular Lines HW

44

7)

8)

9)

10)

45

Day 8 – Equations of Parallel and Perpendicular Lines (5 – 8)

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 = _____


Equation: __________________________________

46

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 = _____


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 = _____


Equation: __________________________________

47

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

48

Challenge

SUMMARY

Exit Ticket

1.

2.

49

Day 8 – Equations of Parallel and Perpendicular Lines - HW

1) Which equation represents a line parallel to the

x-axis?

(a) y 5 (c) x 3

(b) y x 5 (d) x y 3

2) Which equation represents a line parallel to

the line y = 2x – 5?

(a) y = 2x + 5 (c) y = 5x – 2

(b) y = – x – 5 (d) y = –2x – 5

3) Which equation represents a line that is

parallel to the line whose equation

is 2 3 12x y ?

(a) 6 4 2y x (c) 4 6 2x y

(b) 6 4 2y x (d) 6 4 2x y

4) Which equation represents a line that is

parallel to the line y x 3 2 ?

(a) 4 2 5x y (c) y x 3 4

(b) 2 4 1x y (d) y x 4 2

5) Find the equation of the line parallel to the line

whose equation is 2y – 4x = 10 and which passes

through the point (1, 2).

6) Find the equation of the line perpendicular to

the line whose equation is y = 5

6x – 4 and

which passes through the point (5, 3).

50

7) Write an equation of a line that is parallel to

5 15y x and passes through (1, 8).

8) Write an equation of a line that is

perpendicular to 2

65

y x and passes

through (10, –17).

9) Write an equation of a line that is parallel to

2 7y x whose y – intercept is –3.


perpendicular to 3 5y x whose y –

intercept is –3.

11) Write an equation of a line that is parallel to:

29

3y x

.


perpendicular to the line below.

510

6y x

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