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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

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

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 foot-candles 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

-0.75-3

-7.5-9

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