Variation
Variation Functions
Functions
• How do we use solve problems involving
direct, inverse, joint, and combined
variation?
HoltMcDougal
Algebra 2Algebra 2
Holt
Variation Functions
In Chapter 2, you studied many types of linear
functions. One special type of linear function is
called direct variation. A direct variation is a
relationship between two variables x and y that can
be written in the form y = kx, where k ≠ 0. In this
relationship, k is the constant of variation. For
the equation y = kx, y varies directly as x.
Holt McDougal Algebra 2
Variation Functions
A direct variation equation is a linear equation in
the form y = mx + b, where b = 0 and the
constant of variation k is the slope. Because b = 0,
the graph of a direct variation always passes
through the origin.
Holt McDougal Algebra 2
Variation Functions
Example 1: Writing and Graphing Direct Variation
Given: y varies directly as x, and y = 27 when x = 6.
Write and graph the direct variation function.
y = kx
y varies directly as x.
27 = k(6)
Substitute 27 for y and 6 for x.
k = 4.5
Solve for the constant of variation k.
y = 4.5x
Write the variation function by using the
value of k.
Holt McDougal Algebra 2
Variation Functions
Example 1 Continued
y
Graph the direct
variation function.
10
8
y = 4.5x
6
4
The y-intercept is 0
The slope is 4.5
2
–10 –8
–6
–4
–2
–2
–4
–6
–8
–10
Holt McDougal Algebra 2
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4
6
8
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x
Variation Functions
Helpful Hint
If k is positive in a direct variation, the
value of y increases as the value of x
increases.
Holt McDougal Algebra 2
Variation Functions
Example 2: Writing and Graphing Direct Variation
Given: y varies directly as x, and y = 6.5 when x = 13.
Write and graph the direct variation function.
y = kx
6.5 = k(13)
y varies directly as x.
Substitute 6.5 for y and 13 for x.
k = 0.5
Solve for the constant of variation k.
y = 0.5x
Write the variation function by using the
value of k.
Holt McDougal Algebra 2
Variation Functions
Example 2 Continued
y
Graph the direct
variation function.
10
8
y = 0.5x
6
4
The y-intercept is 0
The slope is 0.5
2
–10 –8
–6
–4
–2
–2
–4
–6
–8
–10
Holt McDougal Algebra 2
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Variation Functions
A another type of variation
describes a situation in
which one quantity increases and
the other decreases. For example,
the table shows that the time
needed to drive 600 miles
decreases as speed increases.
This type of variation is an inverse variation. An inverse
variation is a relationship between two variables x and y
k
that can be written in the form y = , where k ≠ 0. For the
x
k
equation y = , y varies inversely as x.
x
Holt McDougal Algebra 2
Variation Functions
Example 3: Writing and Graphing Direct Variation
Given: y varies inversely as x, and y = 4 when x = 5.
Write and graph the inverse variation function.
k
y
x
y varies inversely as x.
k
4
5
Substitute 4 for y and 5 for x.
k  20
Solve for the constant of variation k.
20
y
x
Write the variation function by using the
value of k.
Holt McDougal Algebra 2
Variation Functions
Example 3 Continued
y
Graph the inverse
variation function.
10
8
20
y
x
xy  20
Multiples of 20
6
4
2
–10 –8
–6
–4
–2
–2
–4
–6
–8
–10
Holt McDougal Algebra 2
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4
6
8
10
x
Variation Functions
Example 3: Writing and Graphing Direct Variation
Given: y varies inversely as x, and y = 4 when x = 10.
Write and graph the inverse variation function.
k
y
x
y varies inversely as x.
k
4
10
Substitute 4 for y and 10 for x.
k  40
Solve for the constant of variation k.
40
y
x
Write the variation function by using the
value of k.
Holt McDougal Algebra 2
Variation Functions
Example 3 Continued
y
Graph the inverse
variation function.
40
y
x
xy  40
Multiples of 40
10
8
6
4
2
–10 –8
–6
–4
–2
–2
–4
–6
–8
–10
Holt McDougal Algebra 2
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8
10
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Variation Functions
You can use algebra to rewrite variation
functions in terms of k.
Notice that in direct variation, the ratio of the
two quantities is constant. In inverse variation,
the product of the two quantities is constant.
Holt McDougal Algebra 2
Variation Functions
Example 5: Identifying Direct and Inverse Variation
Determine whether each data set represents a
direct variation, an inverse variation, or neither.
A.
B.
x
y
6.5
8
13 104
4 0.5
x
y
5
30
8
48
Holt McDougal Algebra 2
12
72
In each case xy = 52. The
product is constant, so this
represents an inverse
variation.
y
In each case x = 6. The
ratio is constant, so this
represents a direct variation.
Variation Functions
Example 5: Identifying Direct and Inverse Variation
Determine whether each data set represents a
direct variation, an inverse variation, or neither.
C.
d.
x
y
x
y
3
5
6
14
3.75 15
12
3
Holt McDougal Algebra 2
8
21
5
9
Since xy and y are not
x
constant, this is neither a
direct variation nor an
inverse variation.
In each case xy = 45. The
ratio is constant, so this
represents an inverse
variation.
Variation Functions
Example 5: Identifying Direct and Inverse Variation
Determine whether each data set represents a
direct variation, an inverse variation, or neither.
e.
x
y
1
0.2
Holt McDougal Algebra 2
40
8
26
5.2
y
In each case x = 0.2. The
ratio is constant, so this
represents a direct variation.
Variation Functions
Lesson 6.1 Practice A
Holt McDougal Algebra 2