# Thread: Very Stuck. Help on Squares of functions!

1. ## 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

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

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 $\displaystyle f'(x)$ 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

3. Hello, classicstrings!

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

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

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

4. 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.

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