Step by step
The question
Question: graph f(x) = (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 −39: 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 same power on top and bottom.
The x's cancel, and the numbers in front give 1.
Look: at x = 100, f(x) is about 0.96.
Close to 1.
So the horizontal asymptote is y = 1.
3 Find the intercepts
The graph crosses the x-axis where the top is 0.
The top is 0 at x = 3.
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 = 1 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 7/3.
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.