Graph of Rational Functions

Graph of Rational Functions

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

Graph: y = 1/x

The graph of y = 1/x

This is the graph of y = 1/x.

The graph is at the right top and left bottom of the axes.

Asymptotes

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

The asymptotes are the lines
that the graph follows.

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

For example,
the asymptotes of y = 1/x are
x = 0 and y = 0.

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

The left side cannot be 0.
(The right side's numerator is not 0.)
So y = 0 is the asymptote.

Graph: y = -1/x

The graph of y = -1/x

This is the graph of y = -1/x.

The asymptotes are x = 0 and y = 0.
(the same as y = 1/x)

But the graph is
at the left top and right bottom of the asymptotes.
(different from y = 1/x)

This is because
the sign of the right side's numerator is (-): -1.

Example 1

A rational function is given below. y = 4/(x - 1) + 2 Find the equations of the asymptotes.

Make the rational function in standard form.

Move +2 to the left side.

Find the asymptotes.

x - 1 = 0, y - 2 = 0

So x = 1, y = 2.

This is the graph of y - 2 = 4/x.

The right side's numerator (4) is (+).

So the graph is
at the right top and left bottom of the asymptotes.

Example 2

A rational function is given below. y = 1/(3 - x) - 1 Find the equations of the asymptotes.

Make the rational function in standard form.

Move -1 to the left side.

Change 1/(3 - x) to -1/(x - 3)
to make the coefficient of x to 1.

Find the asymptotes.

x - 3 = 0, y + 1 = 0

So x = 3, y = -1.

This is the graph of y + 1 = -1/(x - 3).

The right side's numerator (-1) is (-).

So the graph is
at the left top and right bottom of the asymptotes.

Example 3

A rational function is given below. y = (3x - 1)/(x - 2) Find the equations of the asymptotes.

Make the rational function in standard form.

To remove 3x in the right side's numerator,
write 3(x - 2),
and write +3⋅2 to undo the former change (-3⋅2).

Split the fraction into

the factored numerator part, 3(x - 2)/(x - 2),

and the remaining part, +5/(x - 2).

Cancel (x - 2) on both sides.

Move 3 to the left side.

Then the rational function is in standard form.

Find the asymptotes.

x - 2 = 0, y - 3 = 0

So x = 2, y = 3.

This is the graph of y - 3 = 5/(x - 2).

The right side's numerator (5) is (+).

So the graph is
at the right top and left bottom of the asymptotes.

Graph: y = 1/x2

The graph of y = 1/x^2

This is the graph of y = 1/x2.

Its asymptotes are x = 0 and y = 0.

The graph is
at the right and left top of the asymptotes.

1/x2 is always (+).
So y is always (+).

So the graph is
at the top of the horizontal asymptote (y = 0).