# Precalculus : Rational Functions

## Example Questions

### Example Question #4 : Find Intercepts And Asymptotes

Find the vertical asymptotes of the following rational function.

No vertical asymptotes.

Explanation:

Finding the vertical asymptotes of the rational function  amounts to finding the roots of the denominator, .

It is easy to check, using the quadratic formula,

that the roots, and thus the asymptotes, are .

### Example Question #5 : Find Intercepts And Asymptotes

Find the y-intercept and asymptote, respectively, of the following function, if possible.

Explanation:

Before we start to simplify the problem, it is crucial to immediately identify the domain of this function .

The denominator cannot be zero, since it is undefined to divide numbers by this value.  After simplification, the equation is:

The domain is  and there is a hole at  since there is a removable discontinuity.  There are no asymptotes.

Since it's not possible to substitute  into the original equation, the y-intercept also does not exist.

### Example Question #6 : Find Intercepts And Asymptotes

What is a vertical asymptote of the following function?

Explanation:

To find the vertical asymptote of a function, we set the denominator equal to

With our function, we complete this process.

The denominator is , so we begin:

### Example Question #7 : Find Intercepts And Asymptotes

What is the -intercept of the following function?

There is no -intercept.

Explanation:

The y-intercept of a function is always found by substituting in .

We can go through this process for our function.

### Example Question #8 : Find Intercepts And Asymptotes

Which of these functions has a vertical asymptote of and a slant asymptote of ?

Explanation:

In order for the vertical asymptote to be , we need the denominator to be . This gives us three choices of numerators:

If the slant asymptote is , we will be able to divide our numerator by and get with a remainder.

Dividing the first one gives us with no remainder.

Dividing the last one gives us with a remainder.

The middle numerator would give us what we were after, with a remainder of -17.

### Example Question #9 : Find Intercepts And Asymptotes

Find the zeros and asymptotes for

.

Zero: ; Asymptotes:

Zero: ; Asymptote:

Zero: ; Asymptotes:

Zeros: ; Asymptotes:

Zeros: ; Asymptote:

Zero: ; Asymptote:

Explanation:

To find the information we're looking for, we should factor this equation:

This means that it simplifies to .

When the equation is in the form of a fraction, to find the zero of the function we need to set the numerator equal to zero and solve for the variable.

To find the asymptote of an equation with a fraction we need to set the denominator of the fraction equal to zero and solve for the variable.

Therefore our equation has a zero at -3 and an asymptote at -2.

### Example Question #10 : Find Intercepts And Asymptotes

Find the slant and vertical asymptotes for the equation

.

Vertical asymptote: ; Slant asymptote:

Vertical asymptote: ; Slant asymptote:

Vertical asymptote: ; Slant asymptote:

Vertical asymptote: ; Slant asymptote:

Vertical asymptote: ; Slant asymptote:

Vertical asymptote: ; Slant asymptote:

Explanation:

To find the vertical asymptote, just set the denominator equal to 0:

To find the slant asymptote, divide the numerator by the denominator, but ignore any remainder. You can use long division or synthetic division.

The slant asymptote is

.

### Example Question #31 : Rational Functions

Find the slant asymptote for

.

This graph does not have a slant asymptote.

This graph does not have a slant asymptote.

Explanation:

By factoring the numerator, we see that this equation is equivalent to

.

That means that we can simplify this equation to .

That means that isn't the slant asymptote, but the equation itself.

is definitely an asymptote, but a vertical asymptote, not a slant asymptote.

### Example Question #32 : Rational Functions

Find the y-intercept of , if any.

Explanation:

Be careful not to confuse this equation with the linear slope-intercept form. The y-intercept of an equation is the y-value when the x-value is zero.

Substitute the value of  into the equation.

Simplify the equation.

The y-intercept is:

### Example Question #33 : Rational Functions

Find the horizontal asymptote of the function: