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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
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Holt McDougal Algebra 1
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Holt McDougal Algebra 1
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).
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Holt McDougal Algebra 1
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?
__________________________________________________________
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Holt McDougal Algebra 1
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.
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Holt McDougal Algebra 1
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? ___________________________
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Holt McDougal Algebra 1
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
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Holt McDougal Algebra 1
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
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Holt McDougal Algebra 1
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.
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Holt McDougal Algebra 1
________________________ ________________________ ________________________
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
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Holt McDougal Algebra 1
________________________ ________________________ ________________________
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Holt McDougal Algebra 1
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.
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Holt McDougal Algebra 1
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. __________________________________
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Holt McDougal Algebra 1
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)
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Holt McDougal Algebra 1
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Holt McDougal Algebra 1
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.
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Holt McDougal Algebra 1
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
_______________________________________
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Holt McDougal Algebra 1
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