5-5: Direct Variation

What’s Direct Variation?

• Direct variation is a function where y = kx, where k ≠ 0

• The variables y and x are vary directly with each other, where k is the constant of variation

• To put simply:–In a direct variation, when one value

increases, the other also increases. (So in the equation y = kx, when

y increases, x also increases)

What’s Direct Variation?

Identify • The equation is a direct variation when…

- it can be written in the form of y = kx

Example 1

• Is the equation a direct variation? If it is, find the constant of variation.

y 7.5x = 0

y – 7.5x = 0y – 7.5x + 7.5x = 0 + 7.5xy = 7.5x

y = 7.5x • YES, it is a direct variation because it can

be written as the form y = kx, the constant of variation (k) = 7.5

6y = 12x 7y = 3x + 4

Quick Check6y = 12x 7y = 3x + 4

6y = 12x

12

y = 2x

YES, it’s a direct variation, k = 2

7y = 3x + 4

3 4

NO, it’s NOT a direct variation

6 6

y = x 6

7 7

y = x + 7 7

Example 2• Write an equation of the direct variation that

includes the given point.(5,1)

• Start with the function form • Substitute (5,1) with (x,y)• Solve for k• Substitute 1/5 for k

y = kx1 = k(5)k = 1/5y = 1/5x

(4, 16) (3, 2)

Quick Check(4, 16) (3, 2)

y = kx16 = k(4)

4 = ky = 4k

y = kx2 = k(3)

2

2

4 4 3 3

3 = k

y = x 3

Example 3• Tony works at a Pizza store, his pay (n) varies directly

with his hours of work (w). On Saturday, Tony worked for 3 hours at the store, and his hourly pay is 20$. Answer the following questions.

a)Write an equation of direct variation for Tony’s pay and his hours of work.

b) What is Tony’s pay on Saturday?

c)What will the graph of this problem look like?

Example 3 (Answers)

• a) y = kx n = 20$w• b) n = 20$(3) n = 60$• c) The graph will be positive, since in a direct

variation, if one variable increases, the other also increases.

Example 3

• Your distance from lightning varies directly with the time it takes you to hear thunder. If you hear thunder 10 seconds after you see lightning, you are about 2 miles from the lightning. Write an equation for the relationship between time and distance.

• Relate: The distance varies directly with the time. When x = 10, y = 2

• Define: Let x = the # of seconds between seeing lightning and hearing thunder Let y = distance in miles from the lightning

y = kx2 = k(10)

1 1

10 10

5 = k y = x5

Quick Check• If you work for 5 hours, you’ll get $90. Write a direct

variation for the relationship between the number of hours and the amount of money.

• Let x = the number of hours• Let y = the amount of money

y = kx90 = k(5)

18 = k y = 18x 5 5

Example 4

x y-10 -5

-4 2

12 -4

x y y/x-5 -10 -10/-5 = 2

2 -4 -4/2 = 2

-4 12 12/-4 = -3

•For each table, use the ratio y/x to tell whether y varies directly with x. If it does, write an equation for the direct variation

No, the ratio y/x is not the same for all pairs of data

x y y/x7 14

1 2

-4 -8

y/x14/7 = 2

2/1 = 2

-8/-4 = 2

Yes, the constant of variation is 2. The equation is y = 2x

Quick Check

x y y/x7 -21

22 -66

-5 15

y/x-21/7 = -3

-66/22 = -3

15/-5 = -3

Yes, the constant of variation is -3. The equation is y = -3x

THE END

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