4.6
Model Direct Variation
Goal
Your Notes
p Write and graph direct variation equations.
VOCABULARY
Direct variation
Constant of variation
Identify direct variation equations
Example 1
Tell whether the equation represents direct variation.
If so, identify the constant of variation.
a. 4x 1 2y 5 0
b. 22x 1 y 5 3
Solution
To tell whether an equation represents direct variation, try
to rewrite the equation in the form y 5 ax.
a. 4x 1 2y 5 0
Write original equation.
2y 5
Subtract
each side.
y5
from
Simplify.
Because the equation 4x 1 2y 5 0
rewritten in the form y 5 ax, it
direct variation. The constant of variation is
b. 22x 1 y 5 3
y5
.
Write original equation.
13
Add
Because the equation 22x 1 y 5 3
rewritten in the form y 5 ax, it
direct variation.
Copyright © Holt McDougal. All rights reserved.
be
to each side.
be
Lesson 4.6 • Algebra 1 Notetaking Guide
109
4.6
Model Direct Variation
Goal
Your Notes
p Write and graph direct variation equations.
VOCABULARY
Direct variation The relationship of two variables,
x and y, provided y 5 ax and a Þ 0
Constant of variation In y 5 ax, a is called the
constant of variation.
Example 1
Identify direct variation equations
Tell whether the equation represents direct variation.
If so, identify the constant of variation.
a. 4x 1 2y 5 0
b. 22x 1 y 5 3
Solution
To tell whether an equation represents direct variation, try
to rewrite the equation in the form y 5 ax.
a. 4x 1 2y 5 0
2y 5 24x
y 5 22x
Write original equation.
Subtract 4x from
each side.
Simplify.
can
be
Because the equation 4x 1 2y 5 0
rewritten in the form y 5 ax, it
represents
direct variation. The constant of variation is 22 .
b. 22x 1 y 5 3
y 5 2x 1 3
Write original equation.
Add 2x to each side.
Because the equation 22x 1 y 5 3 cannot be
rewritten in the form y 5 ax, it does not represent
direct variation.
Copyright © Holt McDougal. All rights reserved.
Lesson 4.6 • Algebra 1 Notetaking Guide
109
Your Notes
Checkpoint Tell whether the equation represents direct
variation. If so, identify the constant of variation.
1. 3x 1 4y 5 0
2. 5x 1 y 5 1
Graph direct variation equations
Example 2
Graph the direct variation equation.
3
a. y 5 25x
The graph of a
direct variation
equation is a line
with a slope of a
and a y-intercept of
0. This line passes
through the origin.
b. y 5 }
x
5
Solution
a. Plot a point at the origin. The slope is equal to the
constant of variation, or
. Find and plot a second
point, then draw a line through the points.
y
1
23
1
21
21
3
x
23
25
b. Plot a point at the origin. The slope is equal to the
constant of variation, or
. Find and plot a second
point, then draw a line through the points.
y
5
3
1
23
110
Lesson 4.6 • Algebra 1 Notetaking Guide
21
21
1
3
5 x
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Tell whether the equation represents direct
variation. If so, identify the constant of variation.
1. 3x 1 4y 5 0
2. 5x 1 y 5 1
3
yes; 2}
4
no
Graph direct variation equations
Example 2
Graph the direct variation equation.
3
a. y 5 25x
The graph of a
direct variation
equation is a line
with a slope of a
and a y-intercept of
0. This line passes
through the origin.
b. y 5 }
x
5
Solution
a. Plot a point at the origin. The slope is equal to the
constant of variation, or 25 . Find and plot a second
point, then draw a line through the points.
y
1
23
(0, 0)
1
21
21
x
3
25
23
(1, 25)
25
1
b. Plot a point at the origin. The slope is equal to the
3
constant of variation, or }
. Find and plot a second
5
point, then draw a line through the points.
y
5
5
3
(5, 3)
3
1
23
110
Lesson 4.6 • Algebra 1 Notetaking Guide
21
21
1
3
5 x
(0, 0)
Copyright © Holt McDougal. All rights reserved.
Your Notes
Write and use a direct variation equation
Example 3
The graph of a direct variation
equation is shown.
y
3
a. Write the direct variation
equation.
(0, 0)
23
b. Find the value of y when
x 5 80.
21
1
3
x
23
(24, 23)
25
Solution
a. Because y varies directly with x, the equation has the
form y 5 ax. Use the fact that y 5 23 when x 5 24
to find a.
y 5 ax
Write direct variation equation.
5 a(
)
Substitute.
5a
Solve for a.
A direct variation equation that relates x and y is
.
y5
b. When x 5 80, y 5
5
.
Checkpoint Complete the following exercises.
1
x.
3. Graph the direct variation equation y 5 }
2
y
3
1
23
21
21
1
3
x
Homework
4. The graph of a direct variation equation passes
through the point (3, 24). Write the direct variation
equation and find the value of y when x 5 15.
Copyright © Holt McDougal. All rights reserved.
Lesson 4.6 • Algebra 1 Notetaking Guide
111
Your Notes
Write and use a direct variation equation
Example 3
The graph of a direct variation
equation is shown.
a. Write the direct variation
equation.
b. Find the value of y when
x 5 80.
y
3
(0, 0)
23
21
1
3
x
23
(24, 23)
25
Solution
a. Because y varies directly with x, the equation has the
form y 5 ax. Use the fact that y 5 23 when x 5 24
to find a.
y 5 ax
Write direct variation equation.
23 5 a( 24 )
Substitute.
3
4
} 5a
Solve for a.
A direct variation equation that relates x and y is
3
x .
y5 }
4
3
b. When x 5 80, y 5 }
(80) 5 60 .
4
Checkpoint Complete the following exercises.
1
x.
3. Graph the direct variation equation y 5 }
2
y
3
1
2
(2, 1)
1
23
21
21
1
3
x
(0, 0)
Homework
4. The graph of a direct variation equation passes
through the point (3, 24). Write the direct variation
equation and find the value of y when x 5 15.
4
3
y 5 2} x; y 5 220
Copyright © Holt McDougal. All rights reserved.
Lesson 4.6 • Algebra 1 Notetaking Guide
111