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Math Help - Very Stuck. Help on Squares of functions!

  1. #1
    Member classicstrings's Avatar
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    Very Stuck. Help on Squares of functions!

    Hi I need some info on how to Sketch the square of functions.

    Eg (x+6)(x+3)(x-4) Sketch the square and comment on the number of turning points.

    Can someone find a website on this? I have searched google to no avail.

    Thanks
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by classicstrings
    Hi I need some info on how to Sketch the square of functions.

    Eg (x+6)(x+3)(x-4) Sketch the square and comment on the number of turning points.

    Can someone find a website on this? I have searched google to no avail.

    Thanks
    The function f(x)=(x+6)^2(x+3)^2(x-4)^2 is always greater than or
    equal to zero, it is increasing as x \to \pm \infty, and touches the x axis when
    x=4,\ -3 and -6.

    Each of the zeros is a turning point, so it must have an additional turning points
    between them making at least 5 turning points. Also it can have at most
    five turning points (as f'(x) is a quintic and so has five zeros
    which are potential turning points), hence it has exactly 5 turning points.

    This should be sufficient to allow you to sketch the curve.

    RonL
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  3. #3
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    Hello, classicstrings!

    An interesting problem . . . I've never been asked to do this.

    f(x)\,=\,(x+6)(x+3)(x-4)

    Sketch the square and comment on the number of turning points.
    f(x) is a cubic with x-intercepts: -6, -3, 4.
    There are two turning points.
    Code:
                                  |
                                  |                   *
                                  |
                   **             |                  *
             -6 *      *          |                 *
          - - o - + - + - o - + - + - + - + - + - o - - - 
             *           -3*      |             * 4
                             *    |           *
            *                    *|       *
                                  |  **
                                  |
    g(x) \,= \,(x=6)^2(x+3)^2(x-4)^2 has the same x-intercepts but of order two.

    . . The graph is tangent to the x-axis there.

    Also the entire graph is above (or on) the x-axis.
    Code:
                                  |
         *                        |   **               *
                    **            |*     *
          *       *   *          *|        *          *
           *     *     *      *   |           *      *
          - - o*- + - + - o*- + - + - + - + - + -*o - - - 
             -6          -3       |               4
    There are five turnning points.
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  4. #4
    Member classicstrings's Avatar
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    Thanks for your help guys! The diagrams + explanations rock! CHeers!!!

    EDIT: If you read what i posted before forget it!

    EDIT: Soroban do you post on other maths help boards? I think I have seen you elsewhere.
    Last edited by classicstrings; May 30th 2006 at 10:06 AM.
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  5. #5
    Member classicstrings's Avatar
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    Also where would the intercepts of the two lines be? Thanks
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