Section 2.1: Introduction to Graphing

In this section, we will try to motivate the importance of learning graphing principles.  We will do this by demonstrating the limitations of point-plotting and graphing calculators without a reasoned, by-hand estimate of the graph.  We will also begin to use the Java-based  graphing tool, XFunctions.

The Limitations of Plotting Points

We have already used 2-dimensional graphs to represent relations and functions whose domain and range are real numbers, assuming familiarity with the basic notion of plotting points.  To review, we associate an ordered pair, such as (1, 2), with the point in the coordinate plane which is 1 unit horizontally and 2 units vertically from the origin:

Since most functions have infinite number of points in their domain, we would need to plot an infinite number of points to accurately represent the function, which is humanly impossible.  However, we typically circumvent this problem in three ways:

• We compute and plot a few points by hand, "connect-the-dots" and hope that this gives a decent approximation to the true graph.  If we plot enough points and we restrict ourselves to relatively simple formulas, our hope is usually not unreasonable.
• We only show those points which fit in a finite "window", given by intervals on the horizontal and vertical axes (in the above plot, we only show the interval [-2, 2] on both the horizontal and vertical axes), and we use our imagination to envision the graph, if it extends beyond the window.
• We use a graphing calculator or computer to plot a lot of points (and "connect-the-dots").  Since a computer can generally work faster and more accurately, we can plot many more points than by hand, so our results should be even better.

Notice that these techniques require us to:

• Choose appropriate points to plot.
• Choose an appropriate plotting window.
• Understand how the graph behaves outside of the plotting window.

This means that we cannot approach this process blindly, but rather we must understand a good deal about the graph before we plot it.

To illustrate, we will try to construct an accurate plot of y = f(x) = 2x - 1 + x(x² - 1)(x² - 4).  If we are not careful, and we only plotted the points at x = -2, -1, 0, 1, and 2, we would calculate the points:

x	y
-2 -1 0 1 2	-5 -3 -1 1 3

and obtain a plot that looks like:

instead of the more accurate plot:

This was simply a problem of plotting too few points.  

You might think that using a computer to plot more points will solve this problem.  However, sometimes the computer plots too many points.  For example, if you use a computer to plot y = g(x) = 1/(x - 1) + 2, you will probably obtain a graph similar to this:

instead of this:

Here the computer plotted too many points; it tried to plot a point at x = 1, which we know does not make sense, since g(1) = 1/(1 - 1) + 2 = 1/0 + 2 is undefined.  

Another problem arises if we choose an inappropriate plot window.  For example, if you plot y = h(x) = -|x - 20| + 19, using a plot window of [-5, 5] on the horizontal and vertical axes, you will get:

and you might be fooled into believing that the graph is a straight line.  A more informative plot is obtained using a window that includes the point (20, 19):

Notice the similarity of this graph with that of the absolute value function.

By learning some basic principles of graphing and the general features of different collections of functions, we will be able to quickly obtain a rough sketch of a large class of functions by:

• plotting a small number of just the "right" points, and

• knowing how to correctly "connect-the-dots" to obtain an accurate picture of the result.  

This will allow us to: 

• detect when a computer or graphing calculator has given us an incorrect or misleading graph, and, if we wish,

• direct a computer or graphing calculator to give us a better graph.

Moreover, we will then be able to view mathematical formulas with a greater sense of understanding (geometrically and numerically) and not simply as a collection of algebraic symbols.

An Introduction to XFunctions

Since not everyone has a graphing calculator, we will use a Java program (so that you can use it anywhere via the Web) as a graphing tool.  It can either be inserted directly in a Web page, like this:

Or it can be launched from a button like this: 

Using this graphing tool, in the next several sections, we will discover and practice the graphing principles that we will use over-and-over again throughout the remainder of the text.  

Learn how to use XFunctions by working through the following worksheet.

Go to Arithmetic and Graphing

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Send questions or comments to jwicks@northpark.edu

Glossary