Question:

Help with quadratic equations?

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I do not understand these problems in my math packet:

Find an equation of the quadratic function described:

1. The minimum value of f is f(3) = -5 and f(1)=2

2. Its graph is a parabola that is tangent to the x-axis at (4,0) and has y-intercept 6.

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  1. 1.

    The general equation of a parabola with a vertical axis is:

    (x - h)^2 = 4a(y - k)

    with vertex (h, k).

    As (x - h)^2 is a square, its minimum value is 0.

    This is when x = h and y = k.

    Therefore h = 3, k = - 5, and a must be positive.

    The equation is:

    (x - 3)^2 = 4a(y + 5).

    Putting x = 1 and y = 2:

    (- 2)^2 = 4a(7)

    4a = 4 / 7.

    The equation is:

    (x - 3)^2 = (4 / 7)(y + 5).

    2.

    (x - h)^2 = 4a(y - k) as before.

    h = 4, k = 0.

    (x - 4)^2 = 4ay

    When x = 0, y = 6:

    (- 4)^2 = 24a

    4a = 8 / 3.

    The equation is:

    (x - 4)^2 = (8 / 3)y.

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