Worked examples
The question
Question: find the asymptotes and holes of f(x) = (x + 1)/(x − 2).
f(x) = x + 1x − 2
An asymptote is a line the graph gets closer and closer to, but never reaches.
We draw it dashed.
asymptote: a line the graph gets close to, but never reaches
1 Vertical asymptotes
We can never divide by 0.
Where the bottom is 0, the graph shoots up or down.
÷ 0: not allowed
The bottom: x − 2 = 0 when x = 2.
x − 2 = 0 → x = 2
Look: at x = 2.1, the bottom is tiny.
So f(x) is about 31: huge.
x = 2.1: bottom 0.1, f(x) ≈ 31
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.
a big x: only the biggest powers count
The same power on top and bottom.
The x's cancel, and the numbers in front give 1.
f(x) ≈ xx = 1
Look: at x = 100, f(x) is about 1.03.
Close to 1.
x = 100: f(100) ≈ 1.03
So the horizontal asymptote is y = 1.
y = 1
So: x = 2 and y = 1.
= x = 2, y = 1
3 Check
Let's check: put x = 2 in the question's bottom.
It gives 0.
Dividing by 0 is not allowed: an asymptote.
x = 2: bottom = 0 ÷ 0
And a very big x, 1000, gives almost exactly 1.
It works!
x = 1000: f(1000) ≈ 1.003 close to 1
Let's look back.
First: vertical asymptotes.
Then: horizontal asymptote.
Last: check.
▶ Watch it solved
The question
Question: find the asymptotes and holes of f(x) = (x − 1)/(x − 1)(x + 2).
f(x) = x − 1(x − 1)(x + 2)
An asymptote is a line the graph gets closer and closer to, but never reaches.
We draw it dashed.
asymptote: a line the graph gets close to, but never reaches
1 Find the holes
Now look for the same bracket on the top and the bottom.
f(x) = (x − 1)(x − 1)(x + 2)
(x − 1) is on the top and the bottom.
(x − 1)/(x − 1) is 1.
Cross them out.
What is left: 1/(x + 2).
f(x) = 1x + 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.
crossed out → a hole (one missing point)
Careful!
A crossed-out bracket gives a hole, not an asymptote.
The graph only misses one point there.
Careful: not x = 1 is an asymptote ✗
Put x = 1 into what is left: y is 1/3.
So the hole is at (1, 1/3).
= hole: x = 1, y = 13
2 Vertical asymptotes
We can never divide by 0.
Where the bottom is 0, the graph shoots up or down.
÷ 0: not allowed
What is left at the bottom: x + 2 = 0 when x = −2.
x + 2 = 0 → x = −2
Look: at x = −1.9, the bottom is tiny.
So f(x) is about 10: huge.
x = −1.9: bottom 0.1, f(x) ≈ 10
3 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.
a big x: only the biggest powers count
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.
f(x) ≈ xx2
Look: at x = 100, f(x) is about 0.01.
Close to 0.
x = 100: f(100) ≈ 0.01
So the horizontal asymptote is y = 0.
y = 0
So: hole (1, 1/3), x = −2 and y = 0.
= hole (1, 13), x = −2, y = 0
4 Check
Let's check: put x = −2 in the question's bottom.
It gives 0.
Dividing by 0 is not allowed: an asymptote.
x = −2: bottom = 0 ÷ 0
And a very big x, 1000, gives almost exactly 0.
It works!
x = 1000: f(1000) ≈ 0.001 close to 0
Let's look back.
First: find the holes.
Then: vertical asymptotes.
Then: horizontal asymptote.
Last: check.
Answer = hole (1, 13), x = −2, y = 0
▶ Watch it solved
The question
Question: graph f(x) = 3x/(x² − 4).
f(x) = 3xx2 − 4
An asymptote is a line the graph gets closer and closer to, but never reaches.
We draw it dashed.
asymptote: a line the graph gets close to, but never reaches
1 Factor the bottom
The bottom: x² − 4.
bottom : x2 − 4
x² − 4 is x² minus 2 squared.
A square minus a square is the plus bracket times the minus bracket.
a² − b² = (a + b)(a − b)
So the bottom is (x + 2)(x − 2).
= (x + 2)(x − 2)
2 Vertical asymptotes
We can never divide by 0.
Where the bottom is 0, the graph shoots up or down.
÷ 0: not allowed
The bottom: x + 2 = 0 when x = −2.
x + 2 = 0 → x = −2
Look: at x = −1.9, the bottom is tiny.
So f(x) is about 15: huge.
x = −1.9: bottom 0.1, f(x) ≈ 15
And x − 2 = 0 when x = 2.
x − 2 = 0 → x = 2
3 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.
a big x: only the biggest powers count
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.
f(x) ≈ 3xx2
Look: at x = 100, f(x) is about 0.03.
Close to 0.
x = 100: f(100) ≈ 0.03
So the horizontal asymptote is y = 0.
y = 0
4 Find the intercepts
The graph crosses the x-axis where the top is 0.
The top is 0 at x = 0.
x-intercept : (0, 0)
Put x = 0 in: f(0) is 0.
The graph crosses the y-axis at (0, 0).
y-intercept : f(0) = 0
5 Draw the graph
A few points on both sides of each vertical asymptote.
We put each x into f.
Draw the grid.
Draw the asymptotes dashed: x = −2 and x = 2 are 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.
6 Check
Let's check one more point: put x = −5 in the question.
We get −5/7.
f(−5) = 3(−5)(−5)2 − 4 = −1521 = −57
The point at x = −5 is on our curve.
It works!
Let's look back.
First: factor the bottom.
Then: vertical asymptotes.
Then: horizontal asymptote.
Then: find the intercepts.
Then: draw the graph.
Last: check.
Answer y-intercept : f(0) = 0
▶ Watch it solved
