Transcript Aim: What is an inverse variation relationship?

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
Aim: Direct & Indirect Variations
Course: Alg. 2 & Trig.
Direct Variation
If a relationship exists between 2 variables
so that their ratio is constant the
relationship is called a direct variation.
y
As x increases,
y
=
kx
 k or
y increases at a
x
constant rate
Constant of Variation
Ex. As you watch a movie, 24 frames flash by every second.
1
24
2
48
3
4
72
96
5
120
Aim: Direct & Indirect Variations
120
# of Frames
x
y
seconds frames
100
linear
equation
80
40
0 1 2 3 4 5 6
Time (secs.)
y = 24x
y
 24
x
Course: Alg. 2 & Trig.
Direct Variation
p varies directly as t. If p = 42
when t = 7, find p when t = 4
Use a proportion to solve:
42 p

7
4
7p = (42)(4)
7p = 168
p = 24
y
 k
x
or
Constant of
Variation?
k=6
y = kx
Constant of Variation
Aim: Direct & Indirect Variations
Course: Alg. 2 & Trig.
Inverse Variation
If x and y vary inversely, then
= a nonzero constant, k.
k =y
x
xy
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
Aim: Direct & Indirect Variations
Course: Alg. 2 & Trig.
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
2
3
4
5
6
Aim: Direct & Indirect Variations
y
30
20
15
12
10
xy = 60
graph this
relationship
Course: Alg. 2 & Trig.
Inverse Variation
Find x when y = 3, if y varies
inversely as x and x = 4, when y = 16
k =y
x
xy = k
Constant of
Variation
Find the value of k
(4)(16) = 64
x(3) = 64
64
x
 21 1
3
3
Aim: Direct & Indirect Variations
Course: Alg. 2 & Trig.
Graphing an Inverse Variation
xy = 60
x
y
days
workers
2
3
4
5
6
30
20
15
12
10
30
w
o
20
r
k
e
r
10
s
0
10
days
20
30
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.
Aim: Direct & Indirect Variations
Course: Alg. 2 & Trig.
Graphing an Inverse Variation
xy = 60
x
y
2
30
3
20
4
15
5
12
6
10
xy = 60
not valid for this
problem
Aim: Direct & Indirect Variations
x
y
-2
-30
-3
-20
-4
-15
-5
-12
-6Course: Alg.-10
2 & Trig.
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
k = $400
xy = k
(cost per person) x (number of persons) = 400
a.
x(25) = 400
b.
x = 16
Aim: Direct & Indirect Variations
12.50y = 400
y = 32
Course: Alg. 2 & Trig.
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 footcandles 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
Aim: Direct & Indirect Variations
Course: Alg. 2 & Trig.
Model Problem
Draw the graph of xy = -12
lines of symmetry
y = -x
graphing
calculator
12
y
x
y=x
combination of numbers that multiply and give -12
x
-1
-2
-3
-4
-6
y
12
6
4
3
2
Aim: Direct & Indirect Variations
x
y
1
-12
2
-6
3
-4
4
-3
Course: Alg. 2 & Trig.
6
-2
Regents Prep
If x varies inversely with y and x = -4
when y = 30, find x when y = 24.
1. -5
2. -3.2
3. 0.005
4. 180
If x varies directly as x and y = 20 when
x = -4, find x when y = 50.
1. -250
2. -10
Aim: Direct & Indirect Variations
3. -0.8
4. 10
Course: Alg. 2 & Trig.
Model Problem
x varies directly as y. If x = 108
when y = 27, find y when x = 56
Use a proportion to solve:
108
56

27
y
108y = (56)(27) = 1512 y = 14
Based on the
table at right,
does y vary
directly with x?
yes
y = -0.75x
x
-3
1
4
10
12
Aim: Direct & Indirect Variations
y
2.25
-0.75
-3
-7.5
-9
y
k
x
2.25
3
 0.75
1
3
4
 7.5
10
9
12
 0.75
 0.75
 0.75
 0.75
 0.75
Course: Alg. 2 & Trig.