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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