September 6Direct Variation

DIRECT VARIATIONx is directly proportional to y

x varies directly as y

YYXX

Direct Variation• y = kx• k is the constant of

variation• the graph must go

through the origin (0,0) and must be linear!!

• Therefore it must be in y = kx form. The y-intercept is 0

Direct Variation

Example NonExample

y = 3x

y = .5x-1

y = 2/3x

y = 5

y = 4 – 6x

y = 11x

y = 8.7x

Direct VariationEx 1)If x varies directly as y and

x = 12 when y = 3, write an equation that relates x and y.

Start with: y = kx

Fill in x and y: 3 = k(12)

Solve for k: 3 1

12 4k

Re-write equation with the k value: y = ¼ x

Same problem, new ?Ex 1)If x varies directly as y and x = 12 when y = 3, find x when y

= 10.

y = ¼ xFill in NEW x and y: 10 = ¼ (x)

Solve for x: x = 40

Another way to do the last ?

2

2

1

1

y

x

y

x FIRST: what you are

comparing

NEXT: substitute your values correctly

LAST: cross multiply to solve for missing variable.

12

3 10

x

12(10) 3

40

x

x

2) If y varies directly as x, and y = 28 when x = 7, find x when y = 52

write an equation that relates x and y.

28 52:

7

y

x x x = 13

What is the constant of variation? 4The constant of variation is the reduced fraction.

y = 4x

3) If y varies directly as the square of x, and y = 4 when x = 3, find y

when x = 6Use a proportion…..

2 2 2

4:3 6

y y

x y = 16

write an equation that relates x and y.4

9y x

4) A car uses 8 gallons of gasoline to travel 290 miles.

How much gasoline will the car

use to travel 400 miles?

8

290 400

gas

m les

x

i

11.034 gallons

5) In scuba diving the time (t) it takes a diver to ascend safely to the surface varies directly with the depth (d) of the dive. It takes a minimum of 3 minutes from a safe ascent from 12 feet. Write an equation that relates depth (d) and time (t). Then determine the minimum time for a safe ascent from 1000 feet?

3 1

12 4

t

d 4t d 4 1000

250min

t

t

6) z varies directly with x and y.

z = kxyWrite the equation relating x, y and z if x = 2, y = -6 and z = 24.

24 (2)( 6)k

24 12k

2k

2z xy

8.2 Inverse Variation

INVERSE or indirect VARIATIONy is inversely proportional to x

y varies inversely as x

K is the constant of variation or constant of proportionality

ky

x

XX YY

Inverse Variation

• This is a NON-LINEAR

function (it doesn’t look like y=mx+b)

• It doesn’t even get close to (0, 0)

• k is still the constant of variation

Inverse Variation

When you buy a car, as time (t) increases, the value (v) decreases.

tt vv

The constant of variation, k is the amount that it decreases.

t is the age of the car.

v is the value of the car.

t

kv

Write the model that represents this situation.

6) If y varies inversely as x and when y = 12, x = 10.

x

ky 12

10

k 120k

120y

x

7)The intensity of a light “I” received from a source varies inversely with the distance “d” from the source. If the light intensity is 10 ft-candles at 21 feet, what is the light intensity at 12 feet? Write your equation first.

kId

1021

k 2100k

2100

12I 175I ft candles

Work with partners on the WS

HW: finish WS 5

show all work!