**Graphing Rational Functions**

If is a rational function given by
where
and are polynomials, we can use the following information to
sketch the graph of :

**I) Asymptotes**

A) __Vertical Asymptotes__

To find the vertical asymptotes, we can first cancel any common factors in and and then take the vertical lines corresponding to the zeros of the denominator:

The line is a vertical asymptote for the graph of whenever and .

The y-coordinates of points on the graph of get arbitrarily
large (in absolute value) as the graph approaches a vertical asymptote, and

__the graph never crosses a vertical asymptote__.

B) __Horizontal Asymptotes__

We can find the horizontal asymptotes by investigating the behavior of as gets arbitrarily large (with either a plus sign or a minus sign):

1. If , then the line (the x-axis) is the horizontal asymptote for the graph of .

2. If , and and are the coefficients of the highest powers of appearing in and , respectively, then the line is the horizontal asymptote for the graph of .

3. If , then there is no horizontal asymptote for the graph of .

The graph of will approach the horizontal asymptote (when
there is one) as gets arbitrarily large (with either a plus sign or a minus
sign).

To determine if the graph crosses a horizontal asymptote with equation ,

we need to solve the equation .

C) __Slanted Asymptotes__

If , then the graph of has a slanted asymptote; and we can find the slanted asymptote by dividing by :

If

where , then the line is the slanted asymptote.

To determine if the graph crosses a slanted asymptote, we need to solve

the equation or, equivalently, the equation .

Notice that if is a rational function, then its graph cannot have

a) two horizontal asymptotes or

b) both a horizontal asymptote and a slanted asymptote.

**II) Intercepts**

The intercepts correspond to the points where the graph intersects the two coordinate axes:

A) To find the y-intercept, set and solve for ; so the y-intercept is given by .

B) To find the x-intercepts, set and solve for ;
so the x-intercepts are the values of for which
(and ).

**III) Sign Chart for **

The sign of indicates where the graph is above or below the x-axis:

A) Where , the graph of is above the x-axis.

B) Where , the graph of is below the x-axis.

(In calculus, you will use sign charts for the first derivative and the second derivative to get more detailed information about the graph of .)

**Ex 1** If
,

find the asymptotes and intercepts for the graph of , and then use this information and a sign chart for to sketch the graph of .

**Sol** 1) Since is a non-constant polynomial, there are no asymptotes for
its graph. (Here
and .)

2) a) , so the y-intercept is -3.

b) for , , or ; so the x-intercepts are -1,1, and 3.

3) Using the facts that and that the sign of changes at -1 and 3
and does not change at 1, we get the following sign chart for :

The following is a sketch of the graph of :

**Ex 2** If

1) Find the asymptotes for the graph of .

2) Find the intercepts for the graph.

3) Make a sign chart for .

4) Determine if the graph of crosses its horizontal asymptote, and if the graph has symmetry around the origin or the y-axis.

5) Use the above information to sketch the graph of .

**Sol**

1) a) , so the vertical asymptotes are the lines and .

b) Since and have the same degree, the horizontal asymptote is the line .

2) a) , so the y-intercept is 2/9.

b) for or , so the x-intercepts are -1 and 1.

3) Using the facts that and all the exponents are odd, we get
the following sign chart for :

4) a) Setting and solving gives , so or . Therefore there is no solution, so the graph of does not cross the horizontal asymptote.

b) Since , is an even function and therefore its graph is symmetric around the y-axis.

5) Using this information, we get the following graph:

**Ex 3** If

1) Find the asymptotes for the graph of .

2) Find the intercepts for the graph.

3) Make a sign chart for .

4) Determine if the graph of crosses its horizontal asymptote, and if the graph has symmetry around the origin or the y-axis.

5) Use the above information to sketch the graph of .

**Sol**

1) a) Since , the vertical asymptotes are the lines and .

b)Since , the horizontal asymptote is the line (the x-axis).

2) a) , so the y-intercept is 0.

b) when , so the x-intercept is 0.

3) Using the facts that and that the exponents are all odd, we
get the following sign chart for :

4) a) Setting and solving gives , so the graph intersects the horizontal asymptote at .

b) Since , is an odd function and therefore its graph is symmetric about the origin.

5) Using the above information, we get the following graph:

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For each of the following functions,

1) Find the asymptotes for the graph of .

2) Find the intercepts for the graph.

3) Make a sign chart for .

4) Determine if the graph of crosses its horizontal asymptote or slanted asymptote (if there is one), and if the graph has symmetry around the origin or the y-axis.

5) Use the above information to sketch the graph of .

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