review for mastery · direct variation a direct variation is a special type of linear relationship....

Name _______________________________________ Date __________________ Class __________________

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Holt McDougal Algebra 1

Review for Mastery

Identifying Linear Functions

You can determine if a function is linear by its graph, ordered pairs, or equation.

Identify whether the graph represents a linear function.

Step 1: Determine whether the graph is a function.

Every x-value is paired with exactly one y-value; therefore,

the graph is a function. Continue to step 2.

Step 2: Determine whether the graph is a straight line.

Conclusion: Because this graph is a function and a straight

line, this graph represents a linear function.

Identify whether {(4, 3), (6, 4), (8, 6)} represents a linear function.

Step 1: Write the ordered pairs in a table.

Step 2: Find the amount of change in each variable.

Determine if the amounts are constant.

Conclusion: Although the x-values show a constant

change, the y-values do not. Therefore, this set of

ordered pairs does not represent a linear function.

Identify whether the function y 5x 2 is a linear function.

Try to write the equation in standard form (Ax By C).

y 5x 2

5x 5x

5x y 2

Conclusion: Because the function can be written in standard form,

(A 5, B 1, C 2), the function is a linear function.

Tell whether each graph, set of ordered pairs, or equation represents

a linear function. Write yes or no.

1. 2. 3.

________________________ ________________________ ________________________

4. {(3, 5), (2, 8), (1, 12)} 5. 2y 3x

2 6. y 4x 7

________________________ ________________________ ________________________

In standard form, x and y

• have exponents of 1

• are not multiplied together

• are not in denominators, exponents,

or radical signs

x y

9 5

5 10

1 15

Name _______________________________________ Date __________________ Class __________________



Name _______________________________________ Date __________________ Class __________________



Give the domain and range for the graphs below.

7. 8. 9.

________________________ ________________________ ________________________

10. Tyler makes $10 per hour at his job. The function f(x) 10x

gives the amount of money Tyler makes after x hours.

Graph this function and give its domain and range.

_____________________________________________________

Review for Mastery

Using Intercepts continued

You can find the x- and y-intercepts from an equation. Then you can use the

intercepts to graph the equation.

Find the x- and y-intercepts of 4x 2y 8.

To find the x-intercept, substitute 0 for y. To find the y-intercept, substitute 0 for x.

4x 2x 8

4x 2(0) 8

4x 8

4x

4

8

4

x 2

4x 2y 8

4(0) 2y 8

2y 8

2y

2

8

2

y 4

The x-intercept is 2. The y-intercept is 4.

Use the intercepts to graph the line described by 4x 2y 8.

Because the x-intercept is 2,

the point (2, 0) is on the graph.

Because the y-intercept is 4,

the point (0, 4) is on the graph.

Plot (2, 0) and (0, 4).

Name _______________________________________ Date __________________ Class __________________



Draw a line through both points.

Find the x- and y-intercepts.

1.

2.

3.

________________________ ________________________ ________________________

4. The volleyball team is traveling to a game 120 miles away.

Their average speed is 40 mi/h. The graphed line describes

the distance left to travel at any time during the trip. Find the

intercepts. What does each intercept represent?

__________________________________________________________

Name _______________________________________ Date __________________ Class __________________



Use intercepts to graph the line described by each equation.

5. 3x 9y 9 6. 4x 6y 12 7. 2x y 4

Review for Mastery

Rate of Change and Slope

A rate of change is a ratio that compares the amount of change in a

dependent variable to the amount of change in an independent variable.

The table shows the average retail price of peanut butter from 1986 to 1997.

Find the rate of change in cost for each time interval. During which time interval

did the cost increase at the greatest rate?

Step 1: Identify independent and dependent variables.

Year is independent. Cost is dependent.

Step 2: Find the rates of change.

1986 to 1987

change in cost

change in years

1.80 1.60

1987 1986

0.20

1 0.2

1987 to 1989

change in cost

change in years

1.811.80

1989 1987

0.01

2 0.005

1989 to 1992 change in cost 1.94 1.81 0.13

0.043change in years 1992 1989 3

1992 to 1997

change in cost

change in years

1.78 1.94

1997 19920.16

5 0.032

The cost increased at the greatest rate from 1986 to 1987.

The table shows the average retail price of cherries from 1986 to 1991.

Find the rate of change in cost for each time interval.

Year 1986 1988 1989 1991

Cost per lb ($) 1.27 1.63 1.15 2.26

Year 1986 1987 1989 1992 1997

Cost per lb ($) 1.60 1.80 1.81 1.94 1.78

greatest rate of change

This rate of change

is negative. The

price went down

during this time

period.

Name _______________________________________ Date __________________ Class __________________



1. 1986 to 1988

change in cost

change in years

2. 1988 to 1989

change in cost

change in years

3. 1989 to 1991

change in cost

change in years

4. Which time interval showed the greatest rate of change? ___________________________

5. Was the rate of change ever negative? If so, when? ___________________________

Name _______________________________________ Date __________________ Class __________________



Find the slope of each line.

6.

7.

8.

________________________ ________________________ ________________________

9.

10.

11.

________________________ ________________________ ________________________

Review for Mastery

The Slope Formula

You can find the slope of a line from any two ordered pairs. The ordered pairs can

be given to you, or you might need to read them from a table or graph.

Find the slope of the line that contains (1, 3) and (2, 0).

Step 1: Name the ordered pairs. (It does not matter which is first

and which is second.)

(1, 3) (2, 0)

Step 2: Label each number in the ordered pairs.

(1, 3) (2, 0)

(x1, y1) (x2, y2)

Step 3: Substitute the ordered pairs into the slope formula.

m 2 1

2 1

y y

x x

0 3

2 ( 1)

first ordered pair second ordered pair

Name _______________________________________ Date __________________ Class __________________



3

3

1

The slope of the line that contains (1, 3) and (2, 0) is 1.

Find the slope of each linear relationship.

1. 2. 3. The line contains

(5, 2) and (7, 6).

________________________ ________________________ ________________________

x y

4 5

8 3

12 1

16 1

Name _______________________________________ Date __________________ Class __________________



Find the slope of the line described by each equation.

4. 2x 5y 10 5. 4x 2y 8 6. 6x 2y 12

________________________ ________________________ ________________________

7. 8y 4x 32 8. 6y 8x 24 9. 1

2 x 2y 3

________________________ ________________________ ________________________

Review for Mastery

Direct Variation

A direct variation is a special type of linear relationship. It can be written

in the form y kx where k is a nonzero constant called the constant of variation.

You can identify direct variations from equations or from ordered pairs.

Tell whether 2x 4y 0 is a direct

variation. If so, identify the constant of

variation.

First, put the equation in the form y kx.

2x 4y 0

2x 2x Add 2x to each side.

4y 2x

4y

4

2x

4 Divide both sides by 4.

y

1

2x

Because the equation can be written in the

form y kx, it is a direct variation.

The constant of variation is

1

2.

Tell whether each equation or relationship is a direct variation. If so,

identify the constant of variation.

1. x y 7 2. 4x 3y 0 3. 8y 24x

Tell whether the relationship is a direct

variation. If so, identify the constant of

variation.

x 2 4 6

y 1 2 3

If we solve y kx for k, we get:

y kx k

Find k for each ordered pair. This means

find for each ordered pair. If they are the

same, the relationship is a direct variation.

This is a direct variation.

Name _______________________________________ Date __________________ Class __________________



________________________ ________________________ ________________________

4. x 4 2 10

y 2 1 5

5.

x 5 12 8

y 17.5 42 28

6. x 6 8 10

y 8 10 12

Name _______________________________________ Date __________________ Class __________________



________________________ ________________________ ________________________

Name _______________________________________ Date __________________ Class __________________



7. The value of y varies directly with x, and y 8 when x 2.

Find y when x 10. ____________________________

8. The value of y varies directly with x, and y 5 when x 20.

Find y when x 35. ____________________________

9. The cost of electricity to run a personal computer is

about $2.13 per day. Write a direct variation equation

for the electrical cost y of running a computer each

day x. Then graph.

________________________________________

Review for Mastery

Slope-Intercept Form

An equation is in slope-intercept form if it is written as:

y mx b.

A line has a slope of 4 and a y-intercept of 3. Write the equation in

slope-intercept form.

y mx b Substitute the given values for m and b.

y 4x 3

A line has a slope of 2. The ordered pair (3, 1) is on the line. Write the

equation in slope-intercept form.

Step 1: Find the y-intercept.

y mx b

y 2x b Substitute the given value for m.

1 2( 3) b Substitute the given values for x and y.

1 6 b Solve for b.

6 6

5 b

Step 2: Write the equation.

y mx b

y 2x 5 Substitute the given value for m and the value you found for b.

Write the equation that describes each line in slope-intercept form.

1. slope

1

4, y-intercept 3 __________________________________

2. slope 5, y-intercept 0 __________________________________

m is the slope.

b is the y-intercept.

Name _______________________________________ Date __________________ Class __________________



3. slope 7, y-intercept 2 __________________________________

4. slope is 3, ( 4, 6) is on the line. __________________________________

5. slope is

1

2, (2, 8) is on the line. __________________________________

6. slope is 1, (5, 2) is on the line. __________________________________

Name _______________________________________ Date __________________ Class __________________



Write the following equations in slope-intercept form.

7. 5x y 30 8. x y 7 9. 4x 3y 12

________________________ ________________________ ________________________

10. Write 2x y 3 in slope-intercept form.

Then graph the line.

______________________________________________

Review for Mastery

Point-Slope Form

You can graph a line if you know the slope and any point on the line.

Graph the line with slope 2 that contains the point (3, 1).

Step 1: Plot (3, 1).

Step 2: The slope is 2 or

2

1; Count 2 up and

1 right and plot another point.

Step 3: Draw a line connecting the points.

Graph the line with the given slope that contains the given point.

1. slope

2

3; (3, 3) 2. slope

1

2; (2, 4) 3. slope 3; (2, 2)

4. slope

3

2; (1, 2) 5. slope 2; (3, 2) 6. slope

2

3; (2, 4)

Name _______________________________________ Date __________________ Class __________________



Name _______________________________________ Date __________________ Class __________________



Review for Mastery

Point-Slope Form continued

You can write a linear equation in slope-intercept form if you are given the slope and a point

on the line, or if you are given any two points on the line.

Write an equation that describes each line in slope intercept form.

slope = 3, (4, 2) is on the line (10, 1) and (8, 5) are on the line

Step 1: Write the equation in Step 1: Find the slope.

point-slope form.

y – 2 = 3(x – 4) 2 1

2 1

5 1 42

8 10 2

y ym

x x

Step 2: Write the equation in Step 2: Substitute the slope and

slope-intercept form by solving one point into the point-slope form.

for x Then write in slope-intercept form.

y – 2 = 3(x – 4) 1 1( )y y m x x

y – 2 = 3x – 12 y 5 2 x 8

2 2 y 5 2x 16

y = 3x – 10 5 5

y = –2x + 21

Write the equation that describes the line in slope-intercept form.

7. slope −3; (1, 2) is on the line 8. slope

1

4; (8, 3) is on the line

________________________________________ ________________________________________

9. slope 4; (2, 8) is on the line 10. (1, 2) and (3, 12) are on the line

________________________________________ ________________________________________

11. (6, 2) and (2, 2) are on the line 12. (4, 1) and (1, 4) are on the line

________________________________________ ________________________________________

Review for Mastery

Slopes of Parallel and Perpendicular Lines

Two lines are parallel if they lie in the same Two lines are perpendicular if they

intersect to form right angles.

Identify which lines are perpendicular.

If the product of the slopes of two lines is

1, the two lines are perpendicular.

Name _______________________________________ Date __________________ Class __________________



plane and have no points in common. The

lines will never intersect.

Identify which lines are parallel.

y 2x 4; y 3x 4; y 2x 1

If lines have the same slope, but different

y-intercepts, they are parallel lines.

y 2x 4; y 3x 4; y 2x 1

m 2, m 3 m 2

b 4 b 4 b 1

y 2x 4 and y 2x 1 are parallel.

Identify which two lines are parallel. Then graph the

parallel lines.

1. y 4x 2; y 2x 1; y 2x 3

_______________________________________

Identify which two lines are perpendicular. Then graph

the perpendicular lines.

2. y

2

3x 2; y

3

2x 1; y

2

3x 3

_______________________________________

Name _______________________________________ Date __________________ Class __________________



Review for Mastery

Slopes of Parallel and Perpendicular Lines continued

Write an equation in slope-intercept form

for the line that passes through

(2, 4) and is parallel to y 3x 2.

Step 1: Find the slope of the line.

The slope is 3.

Step 2: Write the equation in point-slope

form.

y y1 m(x x1)

y 4 3(x 2)

Step 3: Write the equation in slope-intercept

form.

y 4 3(x 2)

y 4 3x 6

4 4

y 3x 2

Write the slope of a line that is parallel to, and perpendicular to,

the given line.

3. y 6x 3 parallel: ______________________ perpendicular: ______________________

4. y

4

3x 1 parallel: ______________________ perpendicular: ______________________

5. Write an equation in slope-intercept form for the line

that passes through (6, 5) and is parallel to y x 4.

_____________________________

6. Write an equation in slope-intercept form for the line

that passes through (8, 1) and is perpendicular to

y 4x 7.

_____________________________

Write an equation in slope-intercept form

for the line that passes through (2, 5) and

is perpendicular to y

2

3x 2.

Step 1: Find the slope of the line and the

slope for the perpendicular line.

The slope is

2

3. The slope of the

perpendicular line will be

3

2.

Step 2: Write the equation (with the new

slope) in point-slope form.

y y1 m(x x1)

y 5

3

2(x 2)

Step 3: Write the equation in slope-intercept

form.

y 5

3

2(x 2)

y 5

3

2x 3

5 5

y

3

2x 8

review for mastery · direct variation a direct variation is a special type of linear relationship....

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