jmerrill, 05 revised 08 section 31 quadratic functions

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  • Section 31Quadratic FunctionsJMerrill, 05Revised 08

  • Definition of a Quadratic FunctionLet a, b, and c be real numbers with a 0. The function given by f(x) = ax2 + bx + c is called a quadratic function

    Your book calls this another form, but this is the standard form of a quadratic function.

  • Parabolas

    The graph of a quadratic equation is a Parabola. Parabolas occur in many real-life situations All parabolas are symmetric with respect to a line called the axis of symmetry.The point where the axis intersects the parabola is the vertex.

    vertex

  • Characteristics

    Graph of f(x)=ax2, a > 0Domain(- , )Range[0, )Decreasing(- , 0)Increasing(0, )Zero/Root/solution(0,0)OrientationOpens up

  • Characteristics

    Graph of f(x)=ax2, a > 0Domain(- , )Range(-, 0]Decreasing(0, )Increasing(-, 0)Zero/Root/solution(0,0)OrientationOpens down

  • Max/MinA parabola has a maximum or a minimum

    maxmin

  • Vertex FormThe vertex form of a quadratic function is given by: f(x) = a(x h)2 + k, a 0

    In this parabola:the axis of symmetry is x = h The vertex is (h, k)If a > o, the parabola opens upward. If a < 0, the parabola opens downward.

  • ExampleIn the equation f(x) = -2(x 3)2 + 8, the graph:

    Opens downHas a vertex at (3, 8)Axis of Symmetry: x = 3Has zeros at 0 = -2(x 3)2 + 8-8 = -2(x 3)24 = (x 3)22 = x 3or-2 = x 3X = 5x = 1

  • Vertex Form from Standard FormDescribe the graph of f(x) = x2 + 8x + 7In order to do this, you have to complete the square to put the problem in vertex form

    Vertex? (-4, -9)Orientation?Opens Up

  • You DoDescribe the graph of f(x) = x2 - 6x + 7Vertex? (3, -2)Orientation?Opens Up

  • ExampleDescribe the graph of f(x) =2x2 + 8x + 7Vertex? (-2, -1)Orientation?Opens Up

  • You DoDescribe the graph of f(x) =3x2 + 6x + 7

    Vertex? (-1, 4)Orientation?Opens Up

  • Write the vertex form of the equation of the parabola whose vertex is (1,2) and passes through (3, - 6)(h,k) = (1,2)

    Since the parabola passes through (3, -6), we know that f(3) = - 6. So:

  • Finding Minimums/MaximumsIf a > 0, f has a minimum at If a < 0, f has a maximum at Ex: a baseball is hit and the path of the baseball is described by f(x)= -0.0032x2 + x + 3. What is the maximum height reached by the baseball?

    Remember the quadratic model is: ax2+bx+cF(x)= - 0.0032(156.25)2+156.25+3 = 81.125 feet

  • Maximizing Area