1. Describe the relationship between x and y.
Aim: What is an direct variation relationship? What is an inverse variation relationship?
Do Now: Fill in the missing values for the table below: x 8 24 72 ?
216
400 ? y 4 12 36
108 ?
200 ?
Describe the relationship between x and y.
x is twice the value of y
Write an algebraic equation that describes the relationship between x and y.
x = 2y or y =1/2x

2. Direct Variation
If a relationship exists between 2 variables so that their ratio is constant the relationship is called a direct variation.
y = kx
Constant of Variation k or
As x increases,
y increases at a
constant rate
Ex. As you watch a movie, 24 frames flash by every second.
Time (secs.)
# of Frames 40 80
100
120 1 2 3 4 5 6
x seconds
y frames 1 2 3 4 5 24 48 72 96
120
linear
equation 24
y = 24x

3. Direct Variation Constant of Variation?
p varies directly as t. If p = when t = 7, find p when t = 4
Use a proportion to solve:
Constant of Variation?
7p = (42)(4)
7p = 168
k = 6
p = 24
y = kx
Constant of Variation k or

4. If x and y vary inversely, then xy = a nonzero constant, k.
Inverse Variation
If x and y vary inversely, then xy = a nonzero constant, k. x k
= y
xy = k
Constant of Variation
Ex. The number of days (x) needed to
complete a job varies inversely as the
number of workers (y) assigned to a job.
If the job can be completed by 2 workers
in 30 days.
What is the constant of variation? 60
What is the equation that represents this
relationship?
xy = 60

5. How many days would it take 3 workers?
Inverse Variation
Ex. The number of days (x) needed to complete a job varies inversely as the number of workers (y) assigned to a job.
If the job can be completed by 2 workers in 30 days.
xy = 60
How many days would it take 3 workers?
What other combinations of xy also satisfy
this relationship? x y 2 30 3 20 4 15 5 12 6 10
graph this
relationship
xy = 60

6. k xy = k = y x Inverse Variation
Find x when y = 3, if y varies inversely as x and x = 4, when y = 16 x k
= y
xy = k
Constant of Variation
Find the value of k
(4)(16) = 64
x(3) = 64

7. Graphing an Inverse Variation30
xy = 60 x
days y
workers 2 30 3 20 4 15 5 12 6 10
workers 20 10 0
10
20
30
days
The graph of an inverse variation relationship is a hyperbola whose center is the origin.
Note: as the days double (x 2) the number
of workers decreased by its reciprocal, 1/2.

8. Graphing an Inverse Variation
xy = 60 x y 2 30 3 20 4 15 5 12 6 10
xy = 60 x y -2
-30 -3
-20 -4
-15 -5
-12 -6
-10
not valid for this
problem

9. Model Problem
The cost of hiring a bus for a trip to Niagara
Falls is $400. The cost per person (x) varies
inversely as the number of persons (y) who
will go on the trip.
a. find the cost per person if 25 go.
b. find the persons who are going
if the cost per person is $12.50
xy = k
k = $400
(cost per person) x (number of persons) = 400 a.
x(25) = 400 b.
12.50y = 400
x = 16
y = 32

10. General equation of inverse variation
Model Problem
The intensity I of light received from a source
varies inversely as the square of the distance d
from the source. If the light intensity is 4 foot-
candles at 17 feet, find the light intensity at
14 feet. Round your answer to the nearest
100th.
General equation of inverse variation
xy = k
(x - represents I)
(y - represents the square of d)
= k
x • d2 = k
substitute to find
constant of I.V.
4 • 172 = k
= 1156
x • 142 = 1156
x = 5.90 foot candles

11. combination of numbers that multiply and give -12 x y -1 12 -2 6 -3 4
Model Problem
Draw the graph of xy = -12
graphing calculator
lines of symmetry
y = -x
y = x
combination of numbers that multiply and give -12 x y -1 12 -2 6 -3 4 -4 3 -6 2 x y 1
-12 2 -6 3 -4 4 -3 6 -2

12. Regents Prep
If x varies inversely with y and x = -4 when y = 30, find x when y = 24.
If x varies directly as x and y = 20 when x = -4, find x when y = 50.

13. x varies directly as y. If x = 108 when y = 27, find y when x = 56
Model Problem
x varies directly as y. If x = when y = 27, find y when x = 56
Use a proportion to solve:
108y = (56)(27)
= 1512
y = 14
Based on the table at right, does y vary directly with x? y x -3 1 4 10 12
2.25
-0.75
-7.5 -9 y
yes
y = -0.75x