Quadratic FunctionsApplications of ParabolasFinding ZerosSlide 4FactoringWarning!!The Quadratic FormulaSlide 8Slide 9Slide 10Concavity and Quadratic FunctionsApplicationsAssignmentQuadratic FunctionsQuadratic FunctionsLesson 2.6Lesson 2.6Applications of ParabolasApplications of ParabolasSolar rays reflect off a parabolic Solar rays reflect off a parabolic mirror and focus at a pointmirror and focus at a pointThis could make a good This could make a good solar powered cookersolar powered cookerToday we look at functions which describe parabolas.Today we look at functions which describe parabolas.Finding ZerosFinding ZerosOften with quadratic functions!!!!Often with quadratic functions!!!! f(x) = a*x f(x) = a*x22 + bx + c!! + bx + c!! we speak of “finding the zeros”we speak of “finding the zeros”This means we wish to find all This means we wish to find all possible values of x for which!!! possible values of x for which!!! a*x a*x22 + bx + c = 0 + bx + c = 0Finding ZerosFinding ZerosAnother way to say this is that we are Another way to say this is that we are seeking the x-axis intercepts seeking the x-axis intercepts This is shown on the graph belowThis is shown on the graph belowHere we see Here we see twotwo zeros – what other zeros – what other possibilities exist?possibilities exist?FactoringFactoringGiven the function!! xGiven the function!! x2 2 - 2x - 8 = 0- 2x - 8 = 0!!Factor the left side of the equation!!! Factor the left side of the equation!!! (x - 4)(x + 2) = 0 (x - 4)(x + 2) = 0 We know that if the product of two numbers!! We know that if the product of two numbers!! a * b = 0!!!! then either ... a * b = 0!!!! then either ... •a = 0!!!! or a = 0!!!! or •b = 0 b = 0 Thus either Thus either •x - 4 = 0!!! ==> x = 4!!!! or x - 4 = 0!!! ==> x = 4!!!! or •x + 2 = 0!!! ==> x = -2 x + 2 = 0!!! ==> x = -2Warning!!Warning!!Problem ... many (most) quadratic Problem ... many (most) quadratic functions are!NOT easily factored!!!functions are!NOT easily factored!!!!!Example:Example:2( ) 3 7 7f x x x= - -The Quadratic FormulaThe Quadratic Formula!!It is possible to create two functions on your It is possible to create two functions on your calculator to use the quadratic formula. calculator to use the quadratic formula. quad1 (a,b,c)!!!!!!!!!! which uses the!!! -b + ... quad1 (a,b,c)!!!!!!!!!! which uses the!!! -b + ... quad2 (a,b,c)!!!!!!!!!! which uses the!!! -b -quad2 (a,b,c)!!!!!!!!!! which uses the!!! -b -The Quadratic FormulaThe Quadratic FormulaTry it for the quadratic functions Try it for the quadratic functions •4x4x22 - 7x + 3 = 0!!!!!!!!!!!!!!!!!!!!!!!!!! - 7x + 3 = 0!!!!!!!!!!!!!!!!!!!!!!!!!!•6x6x22 - 2x + 5 = 0 - 2x + 5 = 0Click to view Spreadsheet SolutionClick to view Spreadsheet SolutionThe Quadratic FormulaThe Quadratic Formula4x4x22 - 7x + 3 = 0!! - 7x + 3 = 0!!The Quadratic FormulaThe Quadratic FormulaWhy does the second function give Why does the second function give "non-real result?“"non-real result?“•6x6x22 - 2x + 5 = 0 - 2x + 5 = 0Concavity and Quadratic FunctionsConcavity and Quadratic FunctionsQuadratic function graphs as a Quadratic function graphs as a parabolaparabola•Will be either concave upWill be either concave up•Or Concave DownOr Concave DownApplicationsApplicationsConsider a ball thrown into the airConsider a ball thrown into the airIt's height (in feet) given byIt's height (in feet) given byh(t) = 80h(t) = 80tt – 16 – 16t t 22Evaluate and interpret h(2)Evaluate and interpret h(2)Solve the equation h(t) = 80Solve the equation h(t) = 80•Interpret the solutionInterpret the solution•Illustrate solution on a graph of h(t)Illustrate solution on a graph of h(t)AssignmentAssignmentLesson 2.6Lesson 2.6Page 92Page 92Exercises 1 – 31 OddExercises 1 – 31