Wanted: graphing procedure

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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 numbers $x=a$ where $f(x)$ goes to infinity as $x$ goes to $a$ (from the right, left, or both). These are the numbers where the graph of $f$ has a vertical asymptote.
(b): Find the critical numbers (the numbers where $f'(x) = 0$ or $f'(x)$ 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 $f''(x) = 0$ or $f''(x)$ is undefined.
(f): If possible, find the $x$-intercepts, the points corresponding to where $f(x) = 0$. Place these points on your graph.
(g): Compute $f'$ and $f''$.
(h): Analyze end behavior: as $x \to \pm \infty $, what happens to the graph of $f$? 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 $f'$ before computing $f''$. Also, one must compute $f'$ 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 $y$-intercept, this is the point $(0,f(0))$. Place this point on your graph.
(b): Find any vertical asymptotes, these correspond to numbers $x=a$ where $f(x)$ goes to infinity as $x$ goes to $a$ (from the right, left, or both).
(c): If possible, find the $x$-intercepts, the points corresponding to where $f(x) = 0$. Place these points on your graph.
(d): Analyze end behavior: as $x \to \pm \infty $, what happens to the graph of $f$? Does it have horizontal asymptotes, increase or decrease without bound, or have some other kind of behavior?
(e): Compute $f'(x)$ and $f''(x)$.
(f): Find the critical numbers (the numbers where $f'(x) = 0$ or $f'(x)$ 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 $f''(x) = 0$ or $f''(x)$ is undefined.
(i): Identify inflection points and concavity.
(j): Determine an interval that shows all relevant behavior

2025-01-06 19:40:55