Aim: Direct & Indirect Variations Course: Alg. 2 & Trig. Aim: What is an direct variation...

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Aim: Direct & Indirect Variations Course: Alg. 2 & Trig.
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 24 216 ?
y 4 12 36 ? ?
?72
108
400
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.
x seconds
y frames
1
2
3
4
5
24
48
72
96
120
Ex. As you watch a movie, 24 frames flash by every second.
Time (secs.)
# of
Fra
mes
40
80
100
120
1 2 3 4 5 60
y = 24x
y = kx
Constant of Variation
x
yk or
x
y24
linearequation
As x increases,y increases at aconstant rate
Aim: Direct & Indirect Variations 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:
47
42 p
7p = (42)(4)7p = 168p = 24
k = 6
Constant of Variation?
y = kx
Constant of Variation
x
yk or
Aim: Direct & Indirect Variations Course: Alg. 2 & Trig.
Inverse Variation
If x and y vary inversely, then xy = a nonzero constant, k.
xy = kxk = y
Constant of Variation
Ex. The number of days (x) needed tocomplete a job varies inversely as the number of workers (y) assigned to a job.If the job can be completed by 2 workersin 30 days.
What is the equation that represents thisrelationship?
What is the constant of variation? 60
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
What other combinations of xy also satisfythis relationship?
x y
2 30
3 20
4 15
5 12
6 10
How many days would it take 3 workers?
graph thisrelationship
xy = 60
Aim: Direct & Indirect Variations Course: Alg. 2 & Trig.
Inverse Variation
Find x when y = 3, if y varies inversely as x and x = 4, when y = 16
(4)(16) = 64
Find the value of k
xy = kxk = y
Constant of Variation
x(3) = 64
64 121 33x
Aim: Direct & Indirect Variations Course: Alg. 2 & Trig.
Graphing an Inverse Variation
xy = 60
xdays
yworkers
2 30
3 20
4 15
5 12
6 10
The graph of an inverse variation relationship is a hyperbola whose center is the origin.
Note: as the days double (x 2) the numberof workers decreased by its reciprocal, 1/2.
30
20
10
10 20 300
workers
days
Aim: Direct & Indirect Variations Course: Alg. 2 & Trig.
Graphing an Inverse Variation
x y
2 30
3 20
4 15
5 12
6 10
xy = 60
xy = 60
x y
2 30
3 20
4 15
5 12
6 10
not valid for thisproblem
Aim: Direct & Indirect Variations Course: Alg. 2 & Trig.
Model Problem
The cost of hiring a bus for a trip to NiagaraFalls is $400. The cost per person (x) variesinversely as the number of persons (y) whowill 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
x(25) = 400
(cost per person) x (number of persons) = 400
x = 16
a. b. 12.50y = 400
y = 32
Aim: Direct & Indirect Variations Course: Alg. 2 & Trig.
substitute to findconstant of I.V.
Model Problem
The intensity I of light received from a sourcevaries 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 at14 feet. Round your answer to the nearest100th. General equation of inverse variation
xy = k
(x  represents I) (y  represents the square of d) = k
x • d2 = k
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
x y
1 12
2 6
3 4
4 3
6 2
x y
1 12
2 6
3 4
4 3
6 2
y = xy = x
lines of symmetry
y 12x
graphing calculator
combination of numbers that multiply and give 12
Aim: Direct & Indirect Variations Course: Alg. 2 & Trig.
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 3. 0.8 4. 10
Aim: Direct & Indirect Variations Course: Alg. 2 & Trig.
x varies directly as y. If x = 108 when y = 27, find y when x = 56
Use a proportion to solve:
y
56
27
108
108y = (56)(27) = 1512 y = 14
Based on the table at right, does y vary directly with x?
yx
3
14
1012
2.25
0.753
7.59
kx
y
75.03
25.2
75.01
75.0
75.04
3
75.010
5.7
75.012
9
yes
y = 0.75x
y
Model Problem