To graph a rational function,
you find the asymptotes
and the intercepts,
plot a few points, and then sketch in the graph. Once you get the swing
of things, rational functions are actually fairly simple to graph. Let's
work through a few examples.

Graph the following:

First I'll find the vertical
asymptotes, if any, for this rational function. Since I can't graph
where the function doesn't exist, and since the function won't exist
where there would be a zero in the denominator, I'll set the denominator
equal to zero to find any forbidden points:

x
– 1 = 0
x
= 1

So I can't have
x
= 1, and therefore
I have a vertical asymptote there.

I'll dash this
in on my graph:

Next I'll find the horizontal
or slant asymptote. Since the numerator and denominator have the same
degree (they're both linear), the asymptote will be horizontal, not
slant, and the horizontal asymptote will be the result of dividing the
leading coefficients:

y
= ^{2}/_{1} = 2

I'll dash this
in, too:

(If you're not sure of
how I found these asymptotes, then review
the lesson on
asymptotes. You will need to be comfortable with this topic before
you proceed further with graphing rationals.)

Next, I'll find any x-
or y-intercepts.

x
= 0: y = ^{(0 + 5)}/_{(0 – 1)} = ^{5}/_{–1}
= –5
y
= 0: 0 = ^{(2x + 5)}/_{(x – 1)}_{ }0
= 2x + 5
–5 = 2x –2.5 = x

Then the intercepts
are at (0,
–5) and
(–2.5,
0). I'll sketch
these in:

I mostly picked x-values
near the middle of the graph: because of the horizontal asymptote, I
already have a good idea of what the graph does off to the sides. (It
can be a good idea to do a point or two near the ends anyway, as a check
on your work.) Also, since I had no intercepts on the right-hand side
of the vertical asymptote to give me hints as to what was happening
with the graph, I needed more points there to show me what was going
on.

Now I'll plot these
points:

And now I can connect
the dots:

When you draw your graph,
make sure you show the graph continuing off to the sides.

Don't just stop at
a point you've drawn, because this will make it look as though the
graph actually stops at that point.

DON'T
DO THIS!

Warning: Your calculator
may display a misleading graph for a given rational function. When you
graph, you plot some points and then you connect them. Your calculator
does the same thing. But you're smart enough to know not to cross a vertical
asymptote. Your calculator isn't that intelligent.

If
your calculator shades a pixel down at the bottom of the screen,
and then the next computed pixel to be shaded is at the top of the
screen, then the calculator is likely to merrily draw a nice vertical
line connecting the two dots, like this:

For
this reason, you may wish to switch your graphing mode from "line"
to "dot" (or the equivalent setting for your calculator
model; check your owner's manual). In "dot" mode, you'll
get a graph that looks like this:

I've highlighted the disconnected
dots in red. The whole graph is dots, but most of the dots are right next
to each other, so they look like a line. If you keep the graphing mode
as "line", you will need to remember that what the screen displays
may be somewhat incorrect when graphing rationals.