5.3 Analyzing 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 also to help us decide where to look. The most interesting features of a graph are the following:

Features of a Graph 1. The x- and y-intercepts: If y = f(x), find the x-intercept(s) by setting y = 0 and solving for x; find the y-intercept by setting x = 0 and solving for y. 2. Relative and absolute extrema: Use the techniques of Section 5.1 to locate the relative and absolute extrema. 3. Points of inflection: Candidates for points of inflection are given by setting the second derivative equal to zero and solving for x. 4. Behavior near points where the function is not defined: If f(x) is not defined at x = a, consider \displaystyle \lim_{x\to a^-}f(x) and \displaystyle \lim_{x\to a^+}f(x) to see how the graph of f behaves as x approaches a. 5. Behavior at infinity: Consider \displaystyle \lim_{x\to -\infty}f(x) and \displaystyle \lim_{x\to +\infty}f(x) if appropriate, to see how the graph of f behaves far to the left and right. Here is an illustration of a graph showing these features: Note  It is sometimes difficult or impossible to solve all of the equations that come up in Steps 1, 2, and 3 of the above analysis. As a consequence, we might not be able to say exactly where the x-intercept, extrema, or points of inflection are. When this happens, we can use graphing technology to assist us in determining accurate numerical approximations.

Here is an exercise on recognizing these features on a graph:

Take a look at the following graph (each gridline represents one unit). To see the coordinates of a point on the graph, just place the cursor on that point.
Note  All answers you enter should be accurate to 1 decimal place unless otherwise stated.

x-intercept(s):		List them from left to right separated by commas; if none, enter "none".
y-intercept:
Absolute and relative maxima:		List them from leftmost to rightmost as pairs (x,y) separated by commas. If none, enter "none"
Absolute and relative minima:		List them from leftmost to rightmost as pairs (x,y) separated by commas. If none, enter "none"
Points of inflection:		Accurate to ± .5. List them from leftmost to rightmost as pairs (x, y) separated by commas. If none, enter "none"

Points where f(x) is not defined:		List them from left to right separated by commas; if none, enter "none".

\displaystyle \lim_{x\to -1^-}f(x) =	\displaystyle \lim_{x\to -1^+}f(x) =	\displaystyle \lim_{x\to 1^-}f(x) =	\displaystyle \lim_{x\to 1^+}f(x) =
Enter "+infinity" for +∞ and "-infinity" for -∞

Behavior at infinity:
\displaystyle \lim_{x\to -\infty}f(x) =	\displaystyle \lim_{x\to +\infty}f(x) =
Enter "+infinity" for +∞ and "-infinity" for -∞

In the next exercise we are given the equation of the graph only:

Consider the function
f(x) = (x-1)^{2/3} + x-1
Before we can analyze this graph, we need to see it. Enter the formula for f(x), adjust the window settings below as you would in a graphing calculator and press "Graph."

f(x) =     xMin = xMax = yMin = yMax =

Note  All answers below must be exact or accurate to at least 4 decimal places!

x-intercepts(s):		List them from left to right separated by commas; if none, enter "none".
y-intercept:		List them from left to right separated by commas; if none, enter "none".
Relative maxima and minima:		List them from leftmost to rightmost as pairs (x, y) separated by commas. If none, enter "none"
Points of inflection:		List them from leftmost to rightmost as pairs (x, y) separated by commas. If none, enter "none"
Isolated points where f(x) is not defined:		List them from leftmost to rightmost as pairs (x, y) separated by commas. If none, enter "none"

Behavior at infinity:
\displaystyle \lim_{x\to -\infty}f(x) =	\displaystyle \lim_{x\to +\infty}f(x) =	Enter "+infinity" for +∞ and "-infinity" for -∞

Now try some of the exercises in Section 5.3 in Applied Calculus or Section 12.3 in Finite Mathematics and Applied Calculus.