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Math Help - Given the absolute minimum determine the value of a function

  1. #1
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    Given the absolute minimum determine the value of a function

    The graph of a function defined on [-3,3] by
    f(x)= x^2 + ax + b
    and has an absolute minimum at (-1,-3). Determine the value of f(1).

    I'm having trouble figuring this problem out. I know the graph is a parabola. Is the y intercept (b) the absolute minimum? Could someone please explain how to solve this?
    Thank you!
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  2. #2
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    Quote Originally Posted by yzobel View Post
    The graph of a function defined on [-3,3] by
    f(x)= x^2 + ax + b
    and has an absolute minimum at (-1,-3). Determine the value of f(1).

    I'm having trouble figuring this problem out. I know the graph is a parabola. <<<< OK
    Is the y intercept (b) the absolute minimum? Could someone please explain how to solve this?
    Thank you!
    1. The coefficient of x is (+1): Therefore the parabola opens up and the absolute Minimum is at the vertex of the parabola.

    2. Since the absolute minimum is (-3) at x = -1 use the vertex-form of the parabola:

    f(x) = a(x-k)^2+h where a determines the shape of the parabola and (k, h) is the vertex of the parabola.

    3. You should come out with:

    f(x)=1 \cdot (x+1)^2-3~\implies~f(x)=x^2+2x-2

    4. Draw a sketch of the parabola.
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  3. #3
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    Quote Originally Posted by yzobel View Post
    The graph of a function defined on [-3,3] by
    f(x)= x^2 + ax + b
    and has an absolute minimum at (-1,-3). Determine the value of f(1).

    I'm having trouble figuring this problem out. I know the graph is a parabola. Is the y intercept (b) the absolute minimum? Could someone please explain how to solve this?
    Thank you!
    No, the constant "b" is merely f(0).

    You have a calculus and non-calculus way to solve this.

    Calculus: The derivative is the slope of the tangent, which is zero at the minimum.

    Then using the x co-ordinate of the minimum, "a" can be found.
    "b" can be found from the y co-ordinate of the minimum.


    Non-calculus: Use the fact that the graph is symmetrical about the minimum, the line x=-1.

    f(1)=f(-3)\Rightarrow\ 1+a+b=9-3a+b\Rightarrow\ 4a=8\Rightarrow\ a=2

    Then you can get "b" from f(-1)=-3.
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    Thank so much guys!
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