Step by step
The question
Question: graph f(x) = (x² − 1)/(x² − x − 2).
An asymptote is a line the graph gets closer and closer to, but never reaches.
We draw it dashed.
1 Factor the top and bottom
The top: x² − 1.
x² − 1 is x² minus 1 squared.
A square minus a square is the plus bracket times the minus bracket.
So the top is (x + 1)(x − 1).
The bottom: x² − x − 2.
We need two numbers.
They multiply to −2, and they add to −1.
They multiply to a minus number, so one number is plus and one is minus.
They add to a minus number, so the bigger one is minus.
We list the pairs that make 2, smallest first.
We take the smaller from the bigger.
We need 1.
1 and 2 give 1.
Yes!
With the signs: 1 and −2.
So the bottom is (x + 1)(x − 2).
2 Find the holes
Now look for the same bracket on the top and the bottom.
(x + 1) is on the top and the bottom.
(x + 1)/(x + 1) is 1.
Cross them out.
What is left: (x − 1)/(x − 2).
But the x that made the crossed bracket 0 is still not allowed in the question.
The graph just has one missing point there: a hole.
Careful!
A crossed-out bracket gives a hole, not an asymptote.
The graph only misses one point there.
Put x = −1 into what is left: y is 2/3.
So the hole is at (−1, 2/3).
3 Vertical asymptotes
We can never divide by 0.
Where the bottom is 0, the graph shoots up or down.
What is left at the bottom: x − 2 = 0 when x = 2.
Look: at x = 2.1, the bottom is tiny.
So f(x) is about 11: huge.
4 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 1.01.
Close to 1.
So the horizontal asymptote is y = 1.
5 Find the intercepts
The graph crosses the x-axis where the top is 0.
The top is 0 at x = 1.
Put x = 0 in: f(0) is 1/2.
The graph crosses the y-axis at (0, 1/2).
6 Draw the graph
A few points on both sides of x = 2.
We put each x into f.
Draw the grid.
Draw the asymptotes dashed: x = 2 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.
The hole: an open circle at (−1, 2/3).
7 Check
Let's check one more point: put x = −3 in the question.
We get 4/5.
The point at x = −3 is on our curve.
It works!
Let's look back.
First: factor the top and bottom.
Then: find the holes.
Then: vertical asymptotes.
Then: horizontal asymptote.
Then: find the intercepts.
Then: draw the graph.
Last: check.