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

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

Direct Variation Constant of Variation? p varies directly as t. If p = 42 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

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

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

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

Graphing an Inverse Variation 30 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.

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

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

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

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

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

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 = 108 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