Post-Composition and Graphing: Worksheet

In this Worksheet, we will practice a technique of analyzing a function to create a sketch by hand, and checking our results using XFunctions.  As before, we will give step-by-step instructions, with each instruction numbered separately.  You should work with a partner and make sure that both of you work with the program for some amount of time.  As you work through the Worksheet, try to answer the bulleted questions.  As usual, you should give an honest attempt to answer the question before looking at the solutions.

For our first example, we will analyze and graph the function f(x) = -2|x| + 1. We begin by decomposing this as:

  1. Take the input and apply the absolute value function, g(x) = |x|,
  2. multiply the result by 2, then
  3. multiply by -1, and
  4. add 1.

Now we will calculate points on the graph of this function numerically as follows:

  1. Start with a table of points of y = g(x) = |x|,
  2. multiply the outputs by 2, then
  3. multiply by -1, and
  4. add 1.
Notice how this step is based on the graphing principles in the text; we apply the same operations in the same order to the output values of the "core" function, g.  To carry out this step, however, we must construct the original table of points for g.

When we calculate points, it is important to choose the right points to plot.  Remember the examples given in the introduction to this chapter.  By not choosing the correct points to plot, we may obtain a misleading result.  The "right" points are those which display the most important features of the graph.  For example, the most important features of the graph of y = g(x) = |x|:

are:

  1. It has a sharp point at x = 0; this is often called its "vertex".
  2. Its "sides" are straight lines.
  3. The two sides look similar; specifically, the graph is "symmetric" with respect to flips of the x-axis (i.e., around the y-axis).

Note: We will see that quite a large number of functions have special behavior near 0, and display some sort of symmetry.  These features can be clearly displayed by plotting the points at x = -1, 0, 1 and "connecting-the-dots" appropriately.

Finally, we want to "connect-the-dots" to obtain our sketch.  However, in order to do this correctly, we must a rough idea of how the graph should look.  To do this, we again use the graphing principles in the text to envision the geometric effects of these operations on the graph of g(x) = |x|.  Remember, since numerically the outputs of the function are effected, all geometric effects will be in the vertical direction.  

Notice that we can use a very similar analysis to graph the function , based on our knowledge of the graph of :

The most important features of this graph are:

  1. It has a kind of "vertex" at (0, 0), coming almost straight down to that point..
  2. The graph curves downwards as it goes to the right.
  3. It goes "off to infinity" up and to the right, but rising rather slowly.

This time, we should use the inputs x = 0, 1, 4, since these are easily calculated points near the vertex and will display the curvature of the graph.  

You should notice that this technique requires that we already know the general characteristics of the graphs of basic functions (e.g., absolute value, square root, etc.) and what are the "important" points to include in any plot.  That is why a major focus of the text will be to build up a general knowledge of such basic functions.  As we learn any new function, we will discuss the important characteristics of its graph and give a short list of "important" points to plot.  We will then practice graphing expressions obtained by applying arithmetic operations to that function.

As a final example, we will apply this technique to sketch a graph of r(x) = -2ëxû + 1.  This will again illustrate the importance of knowing how the graph of s(x) = ëxû looks and thinking about how the graph is transformed geometrically, in addition to simply plotting points.  Remember the graph of s(x) = ëxû:

The important features of this graph are:

  1. It consists of a series or horizontal lines that are closed on one side and open on the other.
  2. The graph is "symmetric" with respect to a shift up and to the right.

Once we plot the points at x = -1, 0, 1, although we cannot simply "connect-the-dots" (or we would obtain a straight line), if we remember these features, we can sketch the horizontal lines from (-1, -1), (0, 0), and (1, 1), and then extend this pattern to obtain the rest of the graph.  Note: While any three consecutive integer points would work just as well, these numbers are the easiest with which to compute.

To summarize, our strategy for sketching the graph of a function that has been obtained by performing arithmetic a "core" function whose graph we know is to:

  1. Recall the basic shape of the "core" function and a table of "important" points on its graph.
  2. Decompose the original function to determine exactly what arithmetic operations have been applied and in what order.
  3. Create a table which displays:
    1. the sequence of arithmetic operations,
    2. the geometric effect of each step,
    3. the numerical effect of each step, and
    4. the corresponding calculated table of values for the "core" function and each intermediate algebraic expression.
  4. Plot the points computed in your table.
  5. Envision the resulting graph by applying the geometric effects to the graph of the "core" function.
  6. "Connect-the-dots" to achieve that graph.

We will refine this graphing technique in the following sections to allow us to graph even more complex expression, and we will use this technique throughout the remainder of the text to graph a wide variety of important mathematical functions.

Go to Graphing and Linear Functions

Table of Contents Send questions or comments to jwicks@northpark.edu Glossary