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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: x824216? y41236?? ? 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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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 Ex. As you watch a movie, 24 frames flash by every second. Time (secs.) # of Frames y = 24x y = kx Constant of Variation k or 24 linear equation As x increases, y increases at a constant rate

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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: 7p = (42)(4) 7p = 168 p = 24 k = 6 Constant of Variation? y = kx Constant of Variation k or

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Aim: Direct & Indirect Variations Course: Alg. 2 & Trig. Inverse Variation If x and y vary inversely, then xy = a nonzero constant, k. xy = k x k = y 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 equation that represents this relationship? What is the constant of variation?60 xy = 60

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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 satisfy this relationship? xy How many days would it take 3 workers? graph this relationship xy = 60

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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 = k x k = y Constant of Variation x(3) = 64

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Aim: Direct & Indirect Variations Course: Alg. 2 & Trig. Graphing an Inverse Variation xy = 60 x days y workers 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/ workersworkers days

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Aim: Direct & Indirect Variations Course: Alg. 2 & Trig. Graphing an Inverse Variation xy xy = 60 xy not valid for this problem

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Aim: Direct & Indirect Variations Course: Alg. 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 xy = kk = $400 x(25) = 400 (cost per person) x (number of persons) = 400 x = 16 a.b.12.50y = 400 y = 32

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Aim: Direct & Indirect Variations Course: Alg. 2 & Trig. substitute to find constant of I.V. 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 100 th. General equation of inverse variation xy = k (x - represents I) (y - represents the square of d) = k x d 2 = k = k = 1156 x 14 2 = 1156 x = 5.90 foot candles

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Aim: Direct & Indirect Variations Course: Alg. 2 & Trig. Model Problem Draw the graph of xy = -12 xy xy y = xy = -x lines of symmetry graphing calculator combination of numbers that multiply and give -12

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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 = If x varies directly as x and y = 20 when x = -4, find x when y =

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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: 108y = (56)(27)= 1512y = 14 Based on the table at right, does y vary directly with x? y x yes y = -0.75x y Model Problem

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