To sketch a rational function's graph, one step is to determine the sign $(+/-)$ of various intervals. I create intervals separated by the vertical asymptote (VA) and $x$-ints on a number line (since these are where a sign change can occur), and test a point in each interval.
As you know, when you cancel a factor in the denominator eg:$(x-5)$, that is where a hole will be $(x=5)$. You then plug the $5$ into the simplified function to get the corresponding $y$-coordinate of the hole.
Normally, I do not include the $x$-value of the hole on the interval number line, since a sign change can only happen at a hole if the hole is on the $x$-axis. This only happens when your cancelled factor cancel in the denominator, and the subsequent plugging in of that $x$-value leads to a $y$-value of 0. However, the only way that can happen is if the cancelled factor is repeated in the numerator (ie: multiplicity $\gt 1$) For example,
$$R(x)=\dfrac{(x-3)(x-3)}{(x-3)(x+5)}$$
The $(x-3)$ terms cancel, and when you do $f(3)$ you will get a $0$ in the numerator from that remaining $(x-3)$ term. So, can we conclude that the only way a hole can involve a sign change is if its on the $x$-axis, and the cancelled factor must have a multiplicity of $\gt 1$ so that the function "equals" zero when you plug that $x$-value into the simplified version. Otherwise, with a multiplicity of just $1$, there is no way the hole can be on the $x$-axis, and there is no need to test for a sign change at that hole… Yea?
Best Answer
To graph a rational function you need to know where the non-removable singularities are. If you reduce this function to lowest terms, you will remove all the x values where it appears to blow up (have a zero denominator), but actually doesn't. That leaves you real singularities (which are generally called poles) where the function will go to $\pm \infty$.
Probably it's best to start your graphing by showing these asymptotes, and then fill in the rest of the sketch afterwards. Figuring out where R(x) = 0 and is thus crossing the x axis doesn't hurt, but isn't necessary. You can sketch using any known points, which is usually easier than finding the roots.
For example if the numerator is not factored and is of the 5th degree, how will you find the roots? But you can sketch it fine without them.