8-6 Direct Variation
Warm Up
Use the point-slope form of each
equation to identify a point the line
passes through and the slope of the
line.
1. y – 3 = – 1
(x – 9)
7
2. y + 2 = 2(x – 5)
3
3. y – 9 = –2(x + 4)
4. y – 5 = – 1 (x + 7)
4
8-6 Direct Variation
A direct variation is a linear function that can
be written as y = kx, where k is a nonzero
constant called the constant of variation.
8-6 Direct Variation
Reading Math
The constant of variation is also called the
constant of proportionality.
8-6 Direct Variation
Additional Example 1A: Determining Whether a Data
Set Varies Directly
Determine whether the data set shows direct
variation.
8-6 Direct Variation
Additional Example 1A Continued
Method 1 Make a graph that shows the
relationship between Adam’s age and his length.
The graph is
not linear.
8-6 Direct Variation
Additional Example 1A Continued
Method 2 Compare ratios.
22 ? 27
3 = 12
81 81 ≠ 264
264
The ratios are not proportional.
The relationship of the data is not a direct
variation.
8-6 Direct Variation
Additional Example 1B: Determining Whether a Data
Set Varies Directly
Determine whether the data set shows direct
variation.
8-6 Direct Variation
Additional Example 1B Continued
Method 1 Make a graph that shows the
relationship between the number of minutes and
the distance the train travels.
Plot the points.
The points lie in
a straight line.
(0, 0) is included.
8-6 Direct Variation
Additional Example 1B Continued
Method 2 Compare ratios.
25 50 75 100
10 = 20= 30 = 40
The ratio is constant for all
the data.
The ratios are proportional. The relationship is
a direct variation.
8-6 Direct Variation
Check It Out: Example 1A
Determine whether the data sets show direct
variation.
Kyle's Basketball Shots
Distance (ft)
Number of Baskets
20
5
30
3
40
0
8-6 Direct Variation
Check It Out: Example 1B
Determine whether the data sets show direct
variation.
Medication Guidelines
Weight (lb)
60
70
80
Dose (mg)
30
35
40
8-6 Direct Variation
Additional Example 2A: Finding Equations of Direct
Variation
Find each equation of direct variation, given
that y varies directly with x.
y is 54 when x is 6
y = kx
54 = k 6
y varies directly with x.
Substitute for x and y.
9=k
Solve for k.
y = 9x
Substitute 9 for k in the original
equation.
8-6 Direct Variation
Additional Example 2B: Finding Equations of Direct
Variation
x is 12 when y is 15
y = kx
15 = k
5=k
4
y =5
4x
y varies directly with x.
12
Substitute for x and y.
Solve for k.
5
Substitute 4 for k in the original
equation.
8-6 Direct Variation
Check It Out: Example 2A
Find each equation of direct variation, given
that y varies directly with x.
y is 7 when x is 3
8-6 Direct Variation
Check It Out: Example 2B
y is 18 when x is 12
8-6 Direct Variation
Check It Out: Example 3A Continued
cards and retail
retail : 20 = 30 = 40; y = 1 x
cards 200 300 400
10
8-6 Direct Variation
Check It Out: Example 3B Continued
cards and shipping
shipping : 5 ≠ 6 ;
cards
200
300