Just taking a quick glance at your work it all seems good. To find the horizontal asymptotes, take and . You do have plenty of information to sketch the graph. You could take the second derivative though to test for concavity.
OK Here we go....
To Sketch the graph of
1. Find the domain.
The denominator is 0 when so the domain of is all of except for
2. Is the function odd or even.
as is not equal to is neither odd or even
3. The intercepts
The x intercept occurs when
The y intercept occurs at
4. Construct a sign table
Having constructed a sign table (don't know how to create it in LATEX) but gives me the following information to help in the sketching of the graph.
is positive on the interval and negative on
5. The Derivative of
has two fixed points at and
As and the fixed points are located at
After applying the first derivative test is found to be a local minimum and is found to be a local maximum.
Also after constructing the sign table of the function is found to be decreasing on and increasing on
As the denominator is 0 when then the lines and are vertical asymptotes
Now here is the question(s)
1 Is all the above correct ?
2 Does the graph have a horizontal asymptote ?
3 Is there enough information the sketch the graph? (I've had a go but I am unable to reproduce my result here.)
You did a good job . . . one error.
is negative on . . . then positive on
is positive on the intervals: . . . no
and negative on
For horizontal asymptotes, examine: . andDerivative of
has fixed points at:
First derivative test
is found to be a local minimum
and is found to be a local maximum.
Sign table of
The function is found to be decreasing on
and increasing on
Vertical asymptotes are: and
We have: .
Hence, the horizontal asymptote is: . , the x-axis.
There is plenty of information (which you found) for sketching the graph.
I'll let you label the intercepts, asymptotes and critical points.Code::* | *: : | : : * | * : : * | : : *| * : * : |* : * * : | : * * --------------:-----+-----:---*---------------------- * : | : * * : | : * : | : * * : | : : | : *: | :*