Family of Quadratic FunctionsGeneral FormZeros of the QuadraticAxis of SymmetryVertex of the ParabolaSlide 6Slide 7Shifting and StretchingOther Quadratic FormsVertex FormAssignmentFamily of Quadratic FunctionsLesson 3.2General Form•Quadratic functions have the standard formy = ax2 + bx + ca, b, and c are constantsa ≠ 0 (why?)•Quadratic functions graph as a parabolaZeros of the Quadratic•Zeros are where the function crosses the x-axisWhere y = 0•Consider possible numbers of zerosNone (or two complex)None (or two complex)OneOneTwoTwoAxis of Symmetry•Parabolas are symmetric about a vertical axis•For y = ax2 + bx + c the axisof symmetry is at•Given y = 3x2 + 8x What is the axis of symmetry?2bxa-=Vertex of the Parabola•The vertex is the “point” of theparabolaThe minimum value Can also be a maximum•What is the x-value of thevertex?•How can we find the y-value?2bxa-=( )2by f x fa-� �= =� �� �Vertex of the Parabola•Given f(x) = x2 + 2x – 8•What is the x-value of the vertex?•What is the y-value of the vertex?•The vertex is at (-1, -9)212 2 1bxa- -= = =-�( 1) 1 2 9 9f - = - - =-Vertex of the Parabola•Given f(x) = x2 + 2x – 8Graph shows vertex at (-1, -9)•Note calculator’s ability to find vertex (minimum or maximum)Shifting and Stretching•Start with f(x) = x2•Determine the results of transformations___ f(x + a) = x2 + 2ax + a2___ f(x) + a = x2 + a___ a * f(x) = ax2___ f(a*x) = a2x2a) horizontal shiftb) vertical stretch or squeezec) horizontal stretch or squeezed) vertical shifte) none of theseOther Quadratic Forms•Standard formy = ax2 + bx + c•Vertex formy = a (x – h)2 + kThen (h,k) is the vertex•Given f(x) = x2 + 2x – 8Change to vertex formHint, use completing the squareExperiment withGeogebra Quadratic FunctionExperiment withGeogebra Quadratic FunctionVertex Form•Changing to vertex form( )2222 82 8y x xy x xy x= + -= + + - -= - +Add something in to make a perfect square trinomialAdd something in to make a perfect square trinomialSubtract the same amount to keep it even.Subtract the same amount to keep it even.Now create a binomial squaredNow create a binomial squaredThis gives us the ordered pair (h,k)This gives us the ordered pair (h,k)Assignment•Lesson 3.2•Page 116•Exercises 1 – 29
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