Step by step
The question
Question: graph f(x) = 3/(x + 1).
An asymptote is a line the graph gets closer and closer to, but never reaches.
We draw it dashed.
1 Vertical asymptotes
We can never divide by 0.
Where the bottom is 0, the graph shoots up or down.
The bottom: x + 1 = 0 when x = −1.
Look: at x = −0.9, the bottom is tiny.
So f(x) is about 30: huge.
2 Horizontal asymptote
For a very big x, the biggest power on the top and on the bottom is all that counts.
The rest is tiny next to it.
The top has a smaller power than the bottom.
When x is huge, the bottom is much bigger, so f(x) gets close to 0.
Look: at x = 100, f(x) is about 0.03.
Close to 0.
So the horizontal asymptote is y = 0.
3 Find the intercepts
The top that is left has no x.
It is never 0, so the graph never crosses the x-axis.
Put x = 0 in: f(0) is 3.
The graph crosses the y-axis at (0, 3).
4 Draw the graph
A few points on both sides of x = −1.
We put each x into f.
Draw the grid.
Draw the asymptotes dashed: x = −1 is vertical.
y = 0 is flat.
Plot the intercepts and the points.
Join the points with a smooth curve.
It gets close to the dashed lines, but never touches them.
5 Check
Let's check one more point: put x = −4 in the question.
We get −1.
The point at x = −4 is on our curve.
It works!
Let's look back.
First: vertical asymptotes.
Then: horizontal asymptote.
Then: find the intercepts.
Then: draw the graph.
Last: check.