19. Properties of Graphs

a. Overview

In this chapter, we study the properties of the graph of a function as well as the graphs of its first and second derivatives. Knowledge of these properties will allow us to roughly sketch the graph. We will also see how knowledge of these properties for one of the graphs will give us information about the shape of the other two graphs. The properties are collected into three groups.

1. Value & Limit Information
   1. When is the function value zero or positive or negative or approaching positive or negative infinity? This entails the discussion of the \(x\)-intercept, and vertical asymptotes.
   2. Where does the graph cross the \(x\)-axis and the \(y\)-axis? This entails the discussion of the \(x\)-intercept and the \(y\)-intercept.
   3. What happens to the function as \(x\) approaches positive or negative infinity? This entails the discussion of horizontal asymptotes and slant asymptotes.
2. First Derivative Information
   1. Where is the function increasing or decreasing or horizontal or vertical?
   2. Use the First Derivative Test to locate the local minima and maxima?
3. Second Derivative Information
   1. Where is the function concave up or concave down and where does it switch concavity?
   2. Use the Second Derivative Test to locate the local minima and maxima?

Most of this is review. We bring it all together here and then use it to draw a rough graph of a function \(f(x)\). In doing this some of the information will be redundant.

Further, we will look at the three graphs of \(f(x)\), \(f'(x)\) and \(f''(x)\) and investigate what each graph says about the other two graphs.