Finding the function of a Parabola

  1. 1. The graph of a quadratic function (a parabola) has x-intercepts -1 and 3 and a range consisting of all number less than or equal to 4. Determine an expression for the function.



    2. none



    3. I do not know how to use the intercepts and the range and manipulate it in to a function. Can you show me step by step? Thank You!
     
  2. jcsd
  3. The parabola has formula y=A(x+B)2+C
    You got the intercepts i.e y=0, x1=-1, x2=3

    Have you ever learned about Vieta's formulas?

    x1+x2=-b/a
    x1*x2=c/a

    where

    x2+(b/a)x+c/a=0

    You already got the solutions of the quadratic equation.
     
  4. Mark44

    Staff: Mentor

    In general, the graph of a parabola is given by the equation y = ax2 + bx + c, which can also be written as y = a(x - r1)(x - r2). Since you are given the two x-intercepts, you should be able to figure out what r1 and r2 are in the second equation I showed.

    So all you need to do is determine that value of a.

    Since the range is all numbers <= 4, the parabola opens downward. The vertex of a parabola is a point on the parabola that is lower than all others (for a parabola that opens upward) or higher than all others (for one that opens downward). The vertex is always midway between the two x-intercepts, provided that there are two x-intercepts. With this information you should be able to find the coordinates of the vertex, and from them the value of a in the parabola's equation.
     
  5. Ok Thank You!
     
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