Graphing Rational Functions

Graphing Rational Functions

How to graph a rational function by using its asymptotes: graphs, asymptotes, examples, and their solutions.

Graphs: y = a/x

This is the graph of the rational function y = a/x, a > 0. If the numerator of the right side is (+), then the graph is at the right top and the left bottom of the asymptotes.

This is the graph of the rational function
y = a/x (a > 0).

The numerator of the right side (a) is (+).

Then the graph is
at the [right top] and the [left bottom]
of the axes (= asymptotes).

(Asymptotes: the line that the graph follows.
You'll learn about the asymptotes below.)

This is the graph of the rational function y = a/x, a < 0. If the numerator of the right side is (-), then the graph is at the left top and the right bottom of the asymptotes.

This is the graph of the rational function
y = a/x (a < 0).

The numerator of the right side (a) is (-).

Then the graph is
at the [left top] and the [right bottom]
of the axes (= asymptotes).

Asymptotes of y = a/x

The asymptotes of a rational function are found by setting (left side) = 0 and (denominator) = 0. So the asymptotes of y = a/x is x = 0 and y = 0.

The asymptotes of a rational function are found by
setting (denominator) = 0 and (left side) = 0.

So the asymptotes of y = a/x are
[x = 0] and [y = 0].

The denominator cannot be 0.
So [x = 0] is the asymptote.

a is not 0.
So the left side cannot be 0.
So [y = 0] is the other asymptote.

By finding the asymptotes,
you can draw the graph of a rational function.

Example 1: Graph y = 4/(x - 1) + 2

Graph the given rational function on a coordinate plane. y = 4/(x - 1) + 2

Move +2 to the left side.

Then y - 2 = 4/(x - 1).

To find the first asymptote of
y - 2 = 4/(x - 1),

set (denominator) = 0.

Then x - 1 = 0.

So the asymptote is [x = 1].

To find the other asymptote of
y - 2 = 4/(x - 1),

set (left side) = 0.

Then y - 2 = 0.

So the other asymptote is [y = 2].

Graph y - 2 = 4/(x - 1)
on the coordinate plane.

First, lightly draw the asymptotes
[x = 1] and [y = 2].

The numerator of 4/(x - 1) is (+).

So draw two curves
at the [right top] and the [left bottom]
of the asymptotes.

Example 2: Graph y = 1/(3 - x) - 1

Graph the given rational function on a coordinate plane. y = 1/(3 - x) - 1

Change the coefficient of x to 1.

So change 1/(3 - x) to -1/(x - 3).

Move -1 to the left side.

Then y + 1 = -1/(x - 3).

To find the first asymptote of
y + 1 = -1/(x - 3),

set (denominator) = 0.

Then x - 3 = 0.

So the asymptote is [x = 3].

To find the other asymptote of
y + 1 = -1/(x - 3),

set (left side) = 0.

Then y + 1 = 0.

So the other asymptote is [y = -1].

Graph y + 1 = -1/(x - 3)
on the coordinate plane.

First, lightly draw the asymptotes
[x = 3] and [y = -1].

The numerator of -1/(x - 3) is (-).

So draw two curves
at the [left top] and the [right bottom]
of the asymptotes.

Example 3: Graph y = (3x - 1)/(x - 2)

Graph the given rational function on a coordinate plane. y = (3x - 1)/(x - 2)

The numerator of the right side
should only have a constant.

So, to cancel the 3x term,
change 3x to 3(x - 2).
(x - 2) is the denominator.

Then, to undo -2,
write +3⋅2.

Split the right side fraction.

Then y = 3(x - 2)/(x - 2) + (3⋅2 - 1)/(x - 2).

3(x - 2)/(x - 2) = 3

3⋅2 = 6

Move 3 to the left side.

6 - 1 = 5

Then y - 3 = 5/(x - 2).

To find the first asymptote of
y - 3 = 5/(x - 2),

set (denominator) = 0.

Then x - 2 = 0.

So the asymptote is [x = 2].

To find the other asymptote of
y - 3 = 5/(x - 2),

set (left side) = 0.

Then y - 3 = 0.

So the other asymptote is [y = 3].

Graph y - 3 = 5/(x - 2)
on the coordinate plane.

First, lightly draw the asymptotes
[x = 2] and [y = 3].

The numerator of 5/(x - 2) is (+).

So draw two curves
at the [right top] and the [left bottom]
of the asymptotes.