Graphing Rational Functions

There are 2 ways to graph rational functions, the long way and the short way (IANS Method).

 

Either way there are a few basic steps that every question starts with:

 

 

 

If the degree of the numerator is less than the degree of the denominator then the HA is y = 0.

 

If the degree of the numerator is greater than the degree of the denominator then there is no HA but there may be an oblique asymptote (slanting asymptote)

 

If the degree of the numerator is equal to the degree of the denominator then the horizontal asymptote is Y = A/B (A being the leading coefficient of the numerator and B being the leading coefficient of the denominator).

 

 

Example

 

F(x) = 1/(x-2)

 

Vertical Asymptote (VA): x = 2

 

Horizontal Asymptote (HA): y = 0

 

X-Intercepts: None - Note if y is set to zero, the first step would be to multiply both sides by (x-2) and then there would no longer be an x, therefore it is possible to conclude that if there is no “x” in the numerator, there will be no x-intercept.

Y-intercepts: y= (-1/2)

 

Behavior of the Function as it approaches the VA (2) from the positive and negative sides

 

 

From the Positive Side (2+)

 

From the Negative Side (2-)

x

y

 

x

y

2.1

10

 

1.9

-10

2.01

100

 

1.99

-100

2.001

1000

 

1.99911

-1000

 

 

To find out the behavior of the function as it approaches a value, take the closes values (in decimals) and substitute them into the function. Then check if the general trend is if the function is increasing or decreasing as it approaches the asymptote. This way when sketching the graph, the direction is known.

 

Repeat the table for the function approaching +infinity and –infinity.

 

As x approaches infinity

 

As x approaches -infinity

x

y

 

x

y

10

0.125

 

-10

-0.83

100

0.010

 

-100

-0.10

1000

0.001

 

-1000

-0.001

 

 

Now all the important information needed to sketch the graph is found.

 

 

 

If you have any questions, leave them in the comments below.