Tutorial: Analyzing graphs

Adaptive game version

(This topic is also in Section 12.4 in Finite Mathematics and Applied Calculus)

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Resources

Function evaluator and grapher

Excel grapher

Features of graphs

It is easy enough to use graphing technology to draw a graph, but we need to use calculus to understand some of the features we are seeing and tell us where to look, and also to help us draw a curve by hand without technology.

To illustrate this point, take a look at the following graph, which has all kinds of interesting-looking features shown with different colors and markers. Each of the buttons shows a particular feature, as listed below.

Features of the graph of $\bold{y = f(x)}$

x- and y-intercepts : $x$-intercepts: In the equation $y = f(x)$, set $y=0$ and solve for $x$.
$y$-intercepts: In the equation $y = f(x)$, set $x=0$ and solve for $y$.
Maxima and minima : Use the techniques of the %%maxmintut to locate and classify the absolute and relative maxima and minima.
Points of inflection : Use the techniques of the %%inflectiontut to locate and classify the points of inflection.
Behavior near singular points of $\bold{f}$ : If $a$ is a singular point of $f$, consider $\lim_{x \to a^-}f(x)$ and $\lim_{x \to a^+}f(x)$ to see how the graph of $f$ behaves as $x \to a$.
Behavior at infinity : Consider $\lim_{x \to -\infty}f(x)$ and $\lim_{x \to \infty}f(x)$ to see how the graph of $f$ behaves far to the left and right.

Note It is sometimes difficult or impossible to solve all of the equations that come up in Steps 1, 2, and 3. As a consequence, we might not be able to say exactly where the $x$-intercepts, extrema, or points of inflection are. When this happens we can use graphing technology to assist us in determining accurate numerical approximations.

In the following quiz, use the graphs of $f'$ and $f''$ to identify the locations of (sometimes hard to see!) features of a given graph.

Next, you should calculate the location of the important features of a curve you are already given.

Finally, you are given a more complicated function whose features to identify and localize using graphing technology. We suggest you use the Zweig grapher.

Now try the exercises in Section 12.4 in Finite Mathematics and Applied Calculus. or move ahead to the next tutorial by pressing "Next tutorial" on the sidebar.