# Worksheet on Rational Functions

A rational function is the quotient of two polynomial
functions, where the denominator is

not zero:

I. The **vertical asymptotes** are lines which the
graph of the function approaches,

but never touches. They can be found by setting the denominator equal to zero

and solving for x. (See III for exceptions to this procedure.) If the
denominator

cannot equal zero, then there are no vertical asymptotes. The equations of the

vertical asymptotes will always be of the form x = a. The **domain **will consist of

all real numbers **except** those numbers which make the denominator equal zero.

Example: If , then the vertical asymptotes
are x = 3

and x = -3. The domain is .

1. Find the equations of the vertical asymptotes and the domains of each of

the following functions.

II. The **horizontal asymptote** is a line which the
graph of the function approaches, but

never touches as** x gets very large or very small**, i.e., as
. The

graph** may** cross the horizontal asymptote for some real value of x. The equation
of

the horizontal asymptote will always be of the form y = b. Since we are
examining

the function as x gets large without bound, the terms of greatest importance are

those with the largest exponent in both the numerator and denominator. There are

three cases to be considered.

A. If the** degree of the numerator is less than the degree of the
denominator**, then the horizontal asymptote is y = 0.

Example: . Consider the equation (found by

considering only the highest degree term in numerator and

denominator). It reduces to . As x gets large without

bound, the fraction gets smaller and smaller and approaches

0. Thus the equation of the asymptote is y = 0.

B. If the** degree of the numerator equals the degree of
the denominator**,

then the horizontal asymptote is y = the quotient of the coefficients of the

highest degree terms.

Example: . Consider the equation
. Since the

x^{2} terms will cancel, the horizontal asymptote is y = 3.

C. If the degree of the numerator is greater than the degree of the denominator,

then there is no horizontal asymptote, but there is an oblique asymptote.

(See IV).

III. The range can frequently be found by using the horizontal asymptote,
coupled with

the graph. It will be essential to see if the graph crosses the horizontal
asymptote.

A. Referring to the function in IIA, we found the horizontal asymptote to be

y = 0. Can f(x) ever equal 0? If f(x)=0, then the numerator must = 0.

3x = 0=> x = 0 . Therefore, the graph of f(x) crosses its own horizontal

asymptote at (0, 0). Looking at the graph of f(x) on the calculator shows

that the range in this case is .

B. Referring to the function in IIB, we found the horizontal asymptote to be

y = 3. Can f(x) ever equal 3? If is to equal
3, then we

see that 3x^{2} - 3x +18 = 3x^{2} - 3x + 5 . This implies that 18 = 5, which is

impossible. Therefore, the graph of f(x) does not cross the horizontal

asymptote in this case. Looking at the graph on the calculator and using

MINIMUM show that the range is [.74, 3), where the .74 is an

approximation.

2. Find the equation of the horizontal asymptote, the coordinates of the point
where

the graph crosses the horizontal asymptote (if it exists), and the ranges of
each of

the following functions.

III. In some cases there is a **“hole” **in the graph. This
happens when there is a factor

which cancels out of the numerator and denominator.

Example:

There will be a hole (but not an asymptote) at x=4. To
find the

corresponding y-value, substitute 4 in place of x in the reduced form of the

function. In this example, there is a hole at the point
.

3. Find the coordinates of the hole in the graph of
.

IV. If the** degree of the numerator is greater than the degree of the
denominator,**

then there will be an **oblique (or slant) asymptote** instead of a horizontal

asymptote. To find the equation of the oblique asymptote, use long division as

follows. (could use synthetic division)

Example:

Divide only the x into
the 2 x^{2} ; this gives 2x

Multiply the (x-3) by the 2x

Subtract and bring down -3

Repeat above steps until all terms are used

This means that . As x
gets large without bound, the fraction

will get smaller and smaller and approach zero. Therefore, the equation of the

oblique asymptote is y = 2x+5. There are no horizontal asymptote if there is an

oblique asymptote.

4. Find the equation of the oblique asymptote.

V. Find the **y-intercept,** let x=0. Some functions may not have a y-intercept. No

function will ever have more than one y-intercept, since this would cause it tot
fail the

vertical line test for a function.

Example: . If x=0, then y = -4, so the
y-intercept is (0,-4).

5. Find the y-intercept of each of the following functions.

VI. To find the **x-intercept**, let y = 0. The only way a fraction can equal zero
is if its

numerator equals 0, but its denominator does not. A function may have 0, 1, or

more x-intercepts.

Example: . Make sure that

neither of these values causes the denominator to = 0. The x-intercepts are (-1,
0),

and .

6. Find the x-intercepts of each of the following functions.