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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 points $$ where $$ goes to infinity as $$ goes to $$ (from the right, left, or both). These are the points where $$ has a vertical asymptote.
(b)
Find the critical points (the points where $$ or $$ is undefined).
(c)
Identify inflection points and concavity.
(d)
Determine an interval that shows all relevant behavior.
(e)
Find the $$-intercept, this is the point $$. Place this point on your graph.
(f)
Find the candidates for inflection points, the points where $$ or $$ is undefined.
(g)
If possible, find the $$-intercepts, the points where $$. Place these points on your graph.
(h)
Compute $$ and $$.
(i)
Analyze end behavior: as $$, 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 points. 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 are points $$ where $$ goes to infinity as $$ goes to $$ (from the right, left, or both).
(c)
If possible, find the $$-intercepts, the points where $$. 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 points (the points 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 where $$ or $$ is undefined.
(i)
Identify inflection points and concavity.
(j)
Determine an interval that shows all relevant behavior