Presentation on theme: "Describe the relationship between x and y."— Presentation transcript:
1 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:x82472?216400?y41236108?200?Describe the relationship between x and y.x is twice the value of yWrite an algebraic equation that describes the relationship between x and y.x = 2y or y =1/2x
2 Direct VariationIf a relationship exists between 2 variables so that their ratio is constant the relationship is called a direct variation.y = kxConstant of VariationkorAs x increases,y increases at aconstant rateEx. As you watch a movie, 24 frames flash by every second.Time (secs.)# of Frames4080100120123456x secondsy frames1234524487296120linearequation24y = 24x
3 Direct Variation Constant of Variation? p varies directly as t. If p = when t = 7, find p when t = 4Use a proportion to solve:Constant of Variation?7p = (42)(4)7p = 168k = 6p = 24y = kxConstant of Variationkor
4 If x and y vary inversely, then xy = a nonzero constant, k. Inverse VariationIf x and y vary inversely, then xy = a nonzero constant, k.xk= yxy = kConstant of VariationEx. The number of days (x) needed tocomplete a job varies inversely as thenumber of workers (y) assigned to a job.If the job can be completed by 2 workersin 30 days.What is the constant of variation?60What is the equation that represents thisrelationship?xy = 60
5 How many days would it take 3 workers? Inverse VariationEx. 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 = 60How many days would it take 3 workers?What other combinations of xy also satisfythis relationship?xy230320415512610graph thisrelationshipxy = 60
6 k xy = k = y x Inverse Variation Find x when y = 3, if y varies inversely as x and x = 4, when y = 16xk= yxy = kConstant of VariationFind the value of k(4)(16) = 64x(3) = 64
7 Graphing an Inverse Variation 30xy = 60xdaysyworkers230320415512610workers20100102030daysThe 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.
8 Graphing an Inverse Variation xy = 60xy230320415512610xy = 60xy-2-30-3-20-4-15-5-12-6-10not valid for thisproblem
9 Model ProblemThe 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 goingif the cost per person is $12.50xy = kk = $400(cost per person) x (number of persons) = 400a.x(25) = 400b.12.50y = 400x = 16y = 32
10 General equation of inverse variation Model ProblemThe intensity I of light received from a sourcevaries inversely as the square of the distance dfrom 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 variationxy = k(x - represents I)(y - represents the square of d)= kx • d2 = ksubstitute to findconstant of I.V.4 • 172 = k= 1156x • 142 = 1156x = 5.90 foot candles
11 combination of numbers that multiply and give -12 x y -1 12 -2 6 -3 4 Model ProblemDraw the graph of xy = -12graphing calculatorlines of symmetryy = -xy = xcombination of numbers that multiply and give -12xy-112-26-34-43-62xy1-122-63-44-36-2
12 Regents PrepIf x varies inversely with y and x = -4 when y = 30, find x when y = 24.If x varies directly as x and y = 20 when x = -4, find x when y = 50.
13 x varies directly as y. If x = 108 when y = 27, find y when x = 56 Model Problemx varies directly as y. If x = when y = 27, find y when x = 56Use a proportion to solve:108y = (56)(27)= 1512y = 14Based on the table at right, does y vary directly with x?yx-31410122.25-0.75-7.5-9yyesy = -0.75x