Chapter 4 : Common Graphs

We started the process of graphing in the previous chapter. However, since the main focus of that chapter was functions we didn’t graph all that many equations or functions. In this chapter we will now look at graphing a wide variety of equations and functions.

Here is a listing of the topics that we’ll be looking at in this chapter.

Lines, Circles and Piecewise Functions – This section is here only to acknowledge that we’ve already talked about graphing these in a previous chapter.

Parabolas – In this section we will be graphing parabolas. We introduce the vertex and axis of symmetry for a parabola and give a process for graphing parabolas. We also illustrate how to use completing the square to put the parabola into the form \(f(x)=a(x-h)^{2}+k\).

Ellipses – In this section we will graph ellipses. We introduce the standard form of an ellipse and how to use it to quickly graph an ellipse.

Hyperbolas – In this section we will graph hyperbolas. We introduce the standard form of a hyperbola and how to use it to quickly graph a hyperbola.

Miscellaneous Functions – In this section we will graph a couple of common functions that don’t really take all that much work to do but will be needed in later sections. We’ll be looking at the constant function, square root, absolute value and a simple cubic function.

Transformations – In this section we will be looking at vertical and horizontal shifts of graphs as well as reflections of graphs about the \(x\) and \(y\)-axis. Collectively these are often called transformations and if we understand them they can often be used to allow us to quickly graph some fairly complicated functions.

Symmetry – In this section we introduce the idea of symmetry. We discuss symmetry about the x-axis, y-axis and the origin and we give methods for determining what, if any symmetry, a graph will have without having to actually graph the function.

Rational Functions – In this section we will discuss a process for graphing rational functions. We will also introduce the ideas of vertical and horizontal asymptotes as well as how to determine if the graph of a rational function will have them.