Sketching graph of y = x^2 + 4x / (x+2)(x-1)(x+4)

[Sham]

Sketch the graph of the function below:

y = x^2 + 4x / (x+2)(x-1)(x+4)

 

[Denis]

Got no graph paper?

 

[soroban]

Re: Sketching

Hello, Sham!

I must assume that you know the basics of graphing.

Sketch the graph of: \(\displaystyle \L\:y \;= \;\frac{x^2\,+\,4x}{(x\,+\,2)(x\,-\,1)(x\,+\,4)}\)

We have: \(\displaystyle \L\:y\;=\;\frac{x(x\,+\,4)}{(x\,+\,2)(x\,-\,1)(x\,+\,4)}\)

. . which reduces to: \(\displaystyle \L\:y \;=\;\frac{x}{(x+2)(x\,-\,1)}\) provided \(\displaystyle x\,\neq\,-4\)

There is a "hole" at \(\displaystyle x\,=\,-4,\:y\,=\,-\frac{2}{5}\)


The graph of: \(\displaystyle \L\:y\;=\;\frac{x}{(x\,+\,2)(x\,-\,1)}\)

. . Its only intercept is the origin: \(\displaystyle (0,0)\)

. . It has vertical asymptotes: \(\displaystyle x\,=\,-2\) and \(\displaystyle x\,=\,1\)

. . It has a horizontal asymptote: \(\displaystyle y \,=\,0\) (x-axis)


Plot a few points to see where the graph is above or below the x-axis.

The graph looks like this:

                      :           |     :
                      :*          |     :*
                      :           |     :
                      : *         |     : *
                      :  *        |     :   *
                      :    *      |     :       *
           -4         :        *  |     :             *
    - - - - + - - - - + - - + - - * - - + - - - - - - - -
     *      :        -2           |  *  1
            o         :           |   * :
                 *    :           |     :
                   *  :           |    *:
                    * :           |     :
                      :           |     :
                     *:           |     :