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March 30, 2017

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Suppose a parabola f(x) has its vertex at (0, 25) and its zeros at x = -5 and x = 5. Then f(x)equals _____.

A. 25-x^2

B. 1- 1/25 x^2

C. x^2-1

D. x^2-25

  • Integrated Math 1 - ,

    The standard equation of a parabola is :

    y = a x ^ 2 + bx + c


    The vertex of a parabola is the point where the parabola crosses its axis.

    If the coefficient of the x ^ 2 term is positive, the vertex will be the lowest point on the graph, the point at the bottom of the “U”-shape.

    If the coefficient of the x ^ 2 term is negative, the vertex will be the highest point on the graph, the point at the top of the “U”-shape

    But the equation for a parabola can also be written in "vertex form":

    y = a * ( x – h ) ^ 2 + k

    Where point (h, k) is the vertex.

    You can see how this relates to the standard equation by multiplying it out:

    y = a ( x – h ) * ( x – h ) + k

    y = a * ( x ^ 2 - 2 * x * h + h ^ 2 ) + k

    y = a x ^ 2 – 2 * a * h * x + a * h ^ 2 + k

    The coefficient of x here is –2 a h.

    This means that in the standard form ;

    y = a * x ^ 2 + b * x + c

    the expression

    - b / 2 a

    gives the x - coordinate of the vertex.

    In this case - b / 2 a = 0

    that means b = 0 so equation of a parabola is :

    y = a x ^ 2 + c


    for x = 0 y = 25

    25 = a * 0 ^ 2 + c

    25 = c

    c = 25

    for x = - 5 y = 0

    0 = a * ( - 5 ) ^ 2 + c

    0 = 25 a + 25

    - 25 a = 25 Divide both sides by - 25

    a = 25 / - 25

    a = - 1


    Also for x = 5 y = 0

    0 = a * 5 ^ 2 + c

    0 = 25 a + 25

    - 25 a = 25 Divide both sides by - 25

    a = 25 / - 25

    a = - 1


    Equation of this parabola is :

    y = - x ^ 2 + 25

  • Integrated Math 1 - ,

    Answer A.

  • Integrated Math 1 - ,

    If you want to see graph go on:

    rechneronline.de

    In blue rectangle type :

    - x ^ 2 + 25

    Set :

    Range x-axis from - 10 to 10

    Range x-axis from - 10 to 40

    and click option Draw

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