Graphing Rational Functions

Sol 1

1) Since is a non-constant polynomial, there are no asymptotes for its graph.

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

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

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

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

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

Sol 2

1) a) The vertical asymptote is the line .

b) Since and have the same degree and they both have leading coefficient 1, the horizontal asymptote is the line or .

2) a) , so the y-intercept is -1.

b) , so the only x-intercept is -2.

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

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

b) is neither even nor odd, so the graph is not symmetric about the y-axis or the origin.

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

Sol 3 [Compare this example to the previous example.]

1) a) Since for , the only vertical asymptote is the line .

b) Since and have the same degree and they both have leading coefficient 1, the horizontal asymptote is the line or .

2) a) a) , so the y-intercept is -1.

b) Since for , ; so the only x-intercept is -2.

3) Since and both exponents are odd (and is undefined at -1),

we get the following sign chart for :

4) a) Solving gives or or . Since is undefined at -1, though, its graph does not cross the horizontal asymptote.

b) is neither even nor odd, so the graph is not symmetric about the y-axis or the origin.

5) Using the above information, we get the graph shown below:

Sol 4

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 2.

b) , so the only x-intercept is 4.

3) Since and all the exponents are odd,

we get the following sign chart for :

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

b) is neither even nor odd, so the graph is not symmetric about the y-axis or the origin.

5) Using the above information, we get the graph shown below:

Sol 5

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

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

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

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

3) Using the facts that and the sign of changes at 4 and at -4 but does not change at 0,

we get the following sign chart for :

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

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

5) Using the information we have found, we get the following graph:

Sol 6

1) a) , so the only vertical asymptote is the line (the y-axis).

b) Since , there is no horizontal asymptote; but since , there is a slanted asymptote:

Dividing by gives the original equation ,

so the line is the slanted asymptote.

2) a) Since is undefined, there is no y-intercept.

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

3) Since , using the facts that and all the exponents are odd gives the following sign chart for :

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

b) is neither even nor odd, so the graph is not symmetric about the y-axis or the origin.

5) Using the above information, we get the graph shown below:

Sol 7

1) a) Since ,

the line is the only vertical asymptote.

b) Since , there is no horizontal asymptote; but since , there is a slanted asymptote:

Dividing by gives the original equation ,

so the line is the slanted asymptote.

2) a) , so the y-intercept is -1/2.

b) , so ; and therefore the only x-intercept is 1.

3) Using the facts that and the sign of changes at 2 but does not change at 1, we get the following sign chart for :

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

b) is neither even nor odd, so the graph is not symmetric about the y-axis or the origin.

5) Using the above information, we get the graph shown below:

Sol 8

1) a) , so the only vertical asymptote is the line .

b) Since , there is no horizontal asymptote; but since , there is a slanted asymptote:

Dividing by gives , so the line is the slanted asymptote.

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

b) or , so the x-intercepts are -2 and 3.

3) Using the facts that and all the exponents are odd,

we get the following sign chart for :

4) a) Setting gives , so and so . Therefore there is no solution, so the graph of does not intersect the slanted asymptote.

b) is neither even nor odd, so the graph is not symmetric about the y-axis or the origin.

5) Using the above information, we get the graph shown below:

Sol 9

1) a) Since for , the only vertical asymptote is the line .

b) Since and have the same degree, the horizontal asymptote is given by or .

2) a) , so the y-intercept is -1/3.

b) , so the only x-intercept is -1.

3) Since and both exponents are odd (and is undefined at 1),

we get the following sign chart for :

4) a) Setting gives , so and . However, is undefined at 1, so the graph of does not intersect the horizontal asymptote.

b) is neither even nor odd, so the graph is not symmetric about the y-axis or the origin.

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

Sol 10

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

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

2) a) , so the y-intercept is 4.

b) or , so the x-intercepts are 4 and -2.

3) Using the facts that and that all the exponents are odd,

we get the following sign chart for :

4) a) Setting gives , so gives or . Therefore the graph of crosses the horizontal asymptote at the point .

b) is neither even nor odd, so the graph is not symmetric about the y-axis or the origin.

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

Sol 11

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

b) Since , there is no horizontal asymptote; but since , there is a slanted asymptote:

Dividing by gives ,

so the line is the slanted asymptote.

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

b) , so the only x-intercept is 2.

3) Using the facts that and that the sign of changes at -1, 2, and 4,

we get the following sign chart for :

4) a) Setting gives , so and therefore so . Since , the graph of intersects the slanted asymptote at the point .

b) is neither even nor odd, so the graph is not symmetric about the y-axis or the origin.

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

Sol 12

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

b) Since and have the same degree, the horizontal asymptote is given by or .

2) a) , so the y-intercept is 3/5.

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

3) Since and all the exponents are odd,

we get the following sign chart for :

4) a) Setting gives , so and or .

Therefore the graph of intersects the horizontal asymptote at the point .

b) is neither even nor odd, so the graph is not symmetric about the y-axis or the origin.

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