Graphs of rational functions

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Rough graphs of rational functions

Graphing rational functions

There is a somewhat wide variation in the graphs of rational functions.

Here we see the the graphs of four rational functions.

Match the curves $A$ , $B$ , $C$ , and $D$ with the functions $\begin{align*} &\frac{x^2-3x+2}{x-2}, &&\frac{x^2-3x+2}{x+2}, \\ &\frac{x-2}{x^2-3x+2}, &&\frac{x+2}{x^2-3x+2}. \end{align*}$

Consider $\frac {x^2-3x+2}{x-2}$ . This function is undefined only at $x=2$ . Of the curves that we see above, $\answer [given]{D}$ is undefined exactly at $x=2$ .

Now consider $\frac {x^2-3x+2}{x+2}$ . This function is undefined only at $x=-2$ . The only function above that undefined exactly at $x=-2$ is curve $\answer [given]{A}$ .

Now consider $\frac {x-2}{x^2-3x+2}$ . This function is undefined at the roots of $x^2-3x+2 = (x-2)(x-1).$ Hence it is undefined at $x=2$ and $x=1$ . It looks like both curves $B$ and $C$ would work. Distinguishing between these two curves is easy enough if we evaluate at $x=-2$ . Check it out.

$\begin{align*} \eval{\frac{x-2}{x^2-3x+2}}_{x=-2} &= \frac{-2-2}{(-2)^2-3(-2)+2}\\ &= \frac{-4}{4+6+2}\\ &=\frac{-4}{12}. \end{align*}$

Since this is negative, we see that $\frac {x-2}{x^2-3x+2}$ corresponds to curve $\answer [given]{B}$ .

Finally, it must be the case that curve $\answer [given]{C}$ corresponds to $\frac {x+2}{x^2-3x+2}$ . We should note that if this function is evaluated at $x=-2$ , the output is zero, and this corroborates our work above.