## Tutorial: Analyzing graphs

(This topic is also in Section 12.4 in*Finite Mathematics and Applied Calculus*) #[I don't like this new tutorial. Take me back to the older tutorial!][No me gusta este nueve tutorial. ¡Regresame al tutorial más viejo (solo en inglés)!]#

### 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 $\bold{y = f(x)}$**

$x$-intercepts: In the equation $y = f(x)$, set $y=0$ and solve for $x$.*x- and y-intercepts :*

$y$-intercepts: In the equation $y = f(x)$, set $x=0$ and solve for $y$.Use the techniques of the %%maxmintut to locate and classify the absolute and relative maxima and minima.*Maxima and minima :*Use the techniques of the %%inflectiontut to locate and classify the points of inflection.*Points of inflection :*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 near singular points of $\bold{f}$ :*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.*Behavior at infinity :*

**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.

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.*February 2022*

Copyright © 2022 Stefan Waner and Steven R. Costenoble