4.5: Curve Sketching

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Throughout this module, if something does not exist, write DNE in the answer box.

Recap Video

Take a look at the following video which recaps the ideas from the section.

Example Video

Below is a video showing a worked example.

Problems

Sketch the graph of $f(x) = x^4+4x$ .

We will break this into steps:

Asymptotes and intercepts: Does the graph of $f(x)$ have any horizontal asymptotes?
Yes No
Does the graph of $f(x)$ have any vertical asymptotes?
Yes No
The $x$ -intercepts of $f(x)$ , in increasing order, are at $x = \answer {-4^{1/3}},\quad x = \answer {0}.$ The $y$ -intercept of $f(x)$ is at $y = \answer {0}$ .
Increasing/Decreasing and Extrema: We find critical points of . Notice This is defined everywhere, and when . So this is our lone critical point. Using our first derivative test, we can find:
- On $(-\infty ,-1)$ , the function $f(x)$ is increasingdecreasing .
- On $(-1,\infty )$ , the function is increasingdecreasing .
This also tells us that there is a local at $x=-1$ . The coordinate on the graph of $f(x)$ is $(-1,\answer {-3})$ .
Concavity and Inflection Points: Notice This is defined everywhere, and is equal to when . Using our number line, we get that:
- On $(-\infty ,0)$ , the function $f(x)$ is concave updown .
- On $(0,\infty )$ , the function is concave updown .
Is there an inflection point at $x=0$ ?
Yes No

Now, we can put all this information together. Try sketching the graph on your own first. Which of the following is the graph of $f(x)$ ?

Sketch the graph of $\displaystyle f(x) = 1+ \frac {x}{x+2}$ .

We will break this into steps:

Asymptotes and intercepts: The graph has a horizontal asymptote at $y = \answer {2}$ and a vertical asymptote at $x = \answer {-2}$ . The $x$ -intercept of $f(x)$ is at $(\answer {-1},0)$ , and the $y$ -intercept of $f(x)$ is $(0,\answer {1})$ .
Increasing/Decreasing and Extrema: We find critical points of . Notice This is undefined at (this is not, technically, a critical point since this point was not in the domain of , but we still need to consider it). We also get when (if no suchpoint exists, write DNE). Using our first derivative test, we can find:
- On $(-\infty ,-2)$ , the function $f(x)$ is increasingdecreasing .
- On $(-2,\infty )$ , the function is increasingdecreasing .
Concavity and Inflection Points: Notice This is undefined at . We also get when (if no suchpoint exists, write DNE).
- On $(-\infty ,-2)$ , the function $f(x)$ is concave updown .
- On $(-2,\infty )$ , the function is concave updown .
Is there an inflection point at $x=-2$ ?
Yes No

Now, we can put all this information together. Try sketching the graph on your own first. Which of the following is the graph of $f(x)$ ?

Sketch the graph of $f(x) = e^x+e^{-x}$ . Label all asymptotes, local extrema, and inflection points. Which of the following is the graph of $f(x)$ ?