Graphing functions

[cmgonz02]

I have been trying to graph functions but this one stopped me, I have searched my math book and cant find anything on how to approach this problem.

y= (x-1)(x-2) <sup>2
----------------
(X+1)

It says graph and find all x-intercepts and asymptotes.

I am pretty sure the x intercepts are X= 1 and X=2</sup>

 

[JeffM]

I have been trying to graph functions but this one stopped me, I have searched my math book and cant find anything on how to approach this problem.

y= (x-1)(x-2) <sup>2
----------------
(X+1)

It says graph and find all x-intercepts and asymptotes.

I am pretty sure the x intercepts are X= 1 and X=2</sup>

Graphing this is hard without calculus (or a graphing calculator). But it is possible with just algebra. What course are you taking, and what exactly does the problem ask you to do?

What you are dealing with is \(\displaystyle y = \dfrac{(x - 1)(x - 2)^2}{x + 1} = \left(\dfrac{x - 1}{x + 1}\right) * (x - 2)^2\), correct? If so,

you should be quite sure that your answers for the x-intercepts are correct because of the zero-product property. If the product of two or more expressions equals zero, then at least one of the factors must equal zero.

\(\displaystyle \dfrac{x - 1}{x + 1} = 0 \implies x - 1 = 0 \implies x = 1\ and\ (x - 2)^2 = 0 \implies (x - 2) = 0 \implies x = 2.\) Good so far.

Now how do you find the y-intercept? Can there be more than one?

OK so is there any restriction on the domain of x?

Answer these questions and we can proceed.

 

[soroban]

Hello, cmgonz02!

If this problem came from your book,
I'm surprised that the technique wasn't explained!

\(\displaystyle y \:=\:\dfrac{(x-1)(x-2)^2}{x+1}\)

Find all x-intercepts and asymptotes, and graph.

We see that the x-intercepts are \(\displaystyle (1,0)\) and \(\displaystyle (2,0).\)
Moreover, (2,0) is to an "even power"; the graph is tangent to the x-axis there.

We see that the denominator equals zero when \(\displaystyle x = \text{-}1.\)
Hence, \(\displaystyle x = \text{-}1\) is a vertical asymptote.

The numerator is a cubic; the denominator is linear.
There is no horizontal asymptote.
The graph goes to \(\displaystyle +\infty\) at the far right and far left.

The graph looks like this:

                : |
    *          *: |
                : |                *
      *       * : |
        *    *  : |               *
           *    : |    *         *
                : |  *   *     *
  --------------:-+-*-------*----------
               -1 | 1       2
                : |*
                : |
                : *
                : |
                : |
                :*|
                : |

**