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Math Help - Function

  1. #1
    Member princess_anna57's Avatar
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    Smile Function

    Sketch the graph of this function:
    f(x) = (4x - 16)^(-1/2)
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  2. #2
    MHF Contributor arbolis's Avatar
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    First, check out the domain of f and compute the limits of f on its domain.
    Once you've done that, find the derivative of f and study its sign in order to find out the intervals in which f is increasing and decreasing. Then look at the sign of the second derivative of f to find out any possible local extrema. After having done this you have a nice idea of what f is looking to. To add more precision in your graph, calculate some values of f, especially where it has local extrema. If it don't have, then chose some random points where to calculate f. Let us know if you have any problem with the steps to do. Good luck.
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  3. #3
    Super Member

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    Hello, princess_anna57!

    Sketch the graph: .f(x) .= .(4x - 16)^{-}

    We have: . y .= .(4x - 16)^{-} . . y(4x - 16)^{} .= .1

    Square both sides: . y(4x - 16) .= .1 . . 4y(x - 4) .= .1

    And we have: . x - 4 .= .1/(4y) . . x .= .(1/4y) + 4


    The graph of: .x .= .1/(4y) .looks like this:
    Code:
            |*
            |
            | *
            |  *
            |    *
            |         *
            |                 *
        - - + - - - - - - - - - - -
            |                 *
            |         *
            |    *
            |  *
            | *
            |
            |*

    The graph of: .x .= .1/(4y) + 4 .is translated 4 units to the right.

    In the original function, we see that y is always positive.
    . . Hence, we want only the upper "half" of the graph.
    Code:
            |       :*
            |       :
            |       : *
            |       :  *
            |       :    *
            |       :         *
            |       :                 *
        - - + - - - + - - - - - - - - - - -
            |       4
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