Two young mathematicians discuss how to sketch the graphs of functions.

Check out this dialogue between two calculus students (based on a true story):
Devyn
Riley, OK I know how to plot something if I’m given a description.
Riley
Yes, it’s kinda fun right?
Devyn
I know! But now I’m not sure how to get the information I need.
Riley
You know, I’d like to make up a procedure based on all these facts, that would tell me what the graph of any function would look like.
Devyn
Me too! Let’s get to work!
Below is a list of features of a graph of a function.
(a)
Find the numbers where goes to infinity as goes to (from the right, left, or both). These are the numbers where the graph of has a vertical asymptote.
(b)
Find the critical numbers (the numbers where or is undefined).
(c)
Identify inflection points and concavity ofr the graph.
(d)
Determine an interval that shows all relevant behavior.
(e)
Find the candidates for inflection points, the points on the graph corresponding to where or is undefined.
(f)
If possible, find the -intercepts, the points corresponding to where . Place these points on your graph.
(g)
Compute and .
(h)
Analyze end behavior: as , what happens to the graph of ? Does it have horizontal asymptotes, increase or decrease without bound, or have some other kind of behavior?
Use either the first or second derivative test to identify local extrema and/or find the intervals where your function is increasing/decreasing. In what order should we take these steps? For example, one must compute before computing . Also, one must compute before finding the critical numbers. There is more than one correct answer.
Here is one possible answer to this question. Compare it with yours!
(a)
Find the -intercept, this is the point . Place this point on your graph.
(b)
Find any vertical asymptotes, these correspond to numbers where goes to infinity as goes to (from the right, left, or both).
(c)
If possible, find the -intercepts, the points corresponding to where . Place these points on your graph.
(d)
Analyze end behavior: as , what happens to the graph of ? Does it have horizontal asymptotes, increase or decrease without bound, or have some other kind of behavior?
(e)
Compute and .
(f)
Find the critical numbers (the numbers where or is undefined).
(g)
Use either the first or second derivative test to identify local extrema and/or find the intervals where your function is increasing/decreasing.
(h)
Find the candidates for inflection points, the points corresponding to where or is undefined.
(i)
Identify inflection points and concavity.
(j)
Determine an interval that shows all relevant behavior