Very Stuck. Help on Squares of functions!

**classicstrings** · May 30th 2006, 05:16 AM

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

**CaptainBlack** · May 30th 2006, 08:47 AM

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

**Soroban** · May 30th 2006, 08:59 AM

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.

**classicstrings** · May 30th 2006, 09:02 AM

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.

**classicstrings** · May 30th 2006, 09:08 AM

Also where would the intercepts of the two lines be? Thanks

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