Step by step
The question
Question: find the asymptotes of f(x) = (3x² + 2x − 1)/(x + 2).
An asymptote is a line the graph gets closer and closer to.
We draw it dashed.
1 Compare the powers
The top has x²: power 2.
The bottom has x: power 1.
The top power is one bigger.
Then there is no flat asymptote.
Instead, the graph gets close to a slanted line: a slant asymptote.
Dividing finds it.
2 Set it up
We write it like long division with numbers.
x + 2 goes outside, 3x² + 2x − 1 inside.
3 First part of the answer
Divide the first parts: 3x² divided by x is 3x.
We write it on top.
Multiply: 3x times (x + 2) is 3x² + 6x.
We write it under, with a minus: we take it away.
3x² take away 3x² is 0.
That is why we chose 3x.
2x take away 6x is −4x.
Bring down the next part: −1.
4 Second part of the answer
Divide the first parts: −4x divided by x is −4.
We write it on top.
Multiply: −4 times (x + 2) is −4x − 8.
We write it under, with a minus: we take it away.
−4x take away −4x is 0.
That is why we chose −4.
−1 take away −8 is 7.
7 has no x.
We can't divide it by x.
So 7 is the remainder: what is left over.
So f(x) is 3x − 4, plus 7/(x + 2).
5 Read the slant asymptote
When x is big, 7/(x + 2) is tiny.
At x = 100 it is 7/102: about 0.07.
Almost 0.
So for a big x, f(x) is close to 3x − 4.
The slant asymptote is y = 3x − 4.
Careful!
The remainder part is not in the line.
It gets smaller and smaller, so we leave it out.
6 Vertical asymptote
The bottom: x + 2 = 0 when x = −2.
We can never divide by 0.
The top is not 0 there, so near it the graph shoots up or down: a vertical asymptote.
So the asymptotes are y = 3x − 4 and x = −2.
7 Check
Let's check with a picture.
The vertical asymptote x = −2, dashed.
The slant asymptote y = 3x − 4, dashed.
The graph of f: on each side it gets close to both dashed lines.
And with numbers: at x = 100, f is about 296.07, and the line gives 296.
Very close.
It works!
Let's look back.
First: compare the powers.
Then: set it up.
Then: first part of the answer.
Then: second part of the answer.
Then: read the slant asymptote.
Then: vertical asymptote.
Last: check.