Section 2.3: Post-Composition and Graphing

In the previous section, we discussed how basic arithmetic affects the number line.  Now we want to see how post-composition with arithmetic functions (i.e., applying arithmetic operations after a known function) affects the function and its resulting two-dimensional graph.  

For example, given the function, k, that goes through the points:

x y = k(x)
-4
-1
0
1
4		 1
-1
0
1
-3

consider how post-composition with a function, f, affects this function numerically and algebraically.  Algebraically, post-composition gives , that is, the function f is applied after (on the "outside" of) k.  Numerically, computing simply involves applying f to the outputs of k.  

For example, consider post-composition with each of the arithmetic operations from the previous section:

Applying several operations at once is no more difficult; if r(x) = -2x + 1, so (i.e., " take the input, multiply by 2, multiply by -1, and add 1"), then , and:

x y = -2k(x) + 1
-4
-1
0
1
4		 -1
3
1
-1
7
Since we know, from the previous section, how these operations affect the number line, it is a simple matter to predict how these graphs will look.  Since only the output values in each case have been affected, which are plotted vertically, each of these post-compositions should affect the graph vertically, via the vertical axis.  

For example, since "add 1" shifts the number line by 1 unit in the positive direction, the graph of y = k(x) + 1 should look like the graph of y = k(x), but shifted up one unit.  We can use XFunctions to quickly verify this.

  1. Start XFunctions by clicking  on this button: .  
  2. Resize your browser window and position both windows so you can read these instructions and work with XFunctions.  Do not resize the XFunctions window, or you will distort the dimensions of the graphing window.
  3. Select "Multigraph Utility" from the menu at the top of the XFunctions Main Screen.  Enter "k(x)" in the "y = " box, set the horizontal limits to [-4, 4] and the vertical limits to [-4, 4], and hit the "Graph!" button or the "Enter" key.  Notice how it creates a graph of y = k(x) by connecting-the-dots, using the five pre-assigned points.
  4. Now select "No. 2" in the menu in the lower left, enter "k(x) + 1" in the "y = " box, and graph.
Notice that the two graphs do go through the points:
x y = k(x)   x y = k(x) + 1
-4
-1
0
1
4		 1
-1
0
1
-3		 and -4
-1
0
1
4		 2
0
1
2
-2

as predicted, and the second graph is shifted up by 1 unit.

Similarly, since "multiply by 2" stretches the number line by a factor of 2, the graph of y = 2k(x) should look like the graph of y = k(x), but stretched vertically by a factor of 2.  Since XFunctions allows us not only to pre-define functions, but also sets of graphs, we can verify this even more easily.

  1. Close the XFunctions window and use the following applet for the next experiment.  Note: You will probably want to resize your browser window at this point, so that you can see the applet and the subsequent text.

  2. Select "Example: k vs. 2k" from the menu at the top of the applet.  Note: This will be the last item in the menu.
Notice that the two graphs do go through the points:
x y = k(x)   x y = 2k(x)
-4
-1
0
1
4		 1
-1
0
1
-3		 and -4
-1
0
1
4		 2
-2
0
2
-6

as predicted, and the second graph is stretched vertically (away from the x-axis) by a factor of 2.

We can also verify that negation acts as expected.

  1. Use the following applet for the next experiment.  

  2. Select "Example: k vs. -k" from the menu at the top of the applet.  Note: Again, this is the last item in the menu.
You can easily see that the two graphs do go through the points:
x y = k(x)   x y = -k(x)
-4
-1
0
1
4		 1
-1
0
1
-3		 and -4
-1
0
1
4		 -1
1
0
-1
3

as predicted, and the second graph is flipped vertically (around the x-axis).

Finally, even when we apply several operations, we only need to consider the order of composition.  For example, the graph of should look like the graph of y = k(x), but stretched vertically by a factor of 2 (g(x) = 2x), flipped vertically (h(x) = -x), and shifted up by 1 (f(x) = x + 1).  We can see this easily in XFunctions:

  1. This will allow you to compare the graphs at each successive step; simply select each successive example from the top menu.  Note: As usual, these appear at the bottom of the menu.  The last example shows all the graphs at once.

We can see that the outputs of -2k(x) + 1 are -2 times those of k plus 1:

x k(x) 2k(x) -2k(x) -2k(x) + 1
-4
-1
0
1
4		 1
-1
0
1
-3		 2
-2
0
2
-6		 -2
2
0
-2
6		 -1
3
1
-1
7

and how, from a geometric perspective, these operations take the original graph of k(x) and:

Note: Once we have learned how each operation works separately, the most difficult part of this analysis is to correctly decompose the expression in the correct order; that is because the order of composition does not tend to follow the order in which the operations are written algebraically (in fact, since evaluation works "inside-out", it tends to be somewhat backwards of the written order).

To summarize, we have seen that:

Operations done after applying the function (i.e., post-composition) affect the graph vertically.  Specifically:

While we have showed that this principle applies to composition of a function with any other function, since most other functions are much harder to describe geometrically, we will use this graphing principle primarily in these three cases of multiplication and addition with constants.

On the following worksheet, we will introduce a technique to analyze a known function composed with arithmetic operations to create an accurate sketch.

Go to Graphing and Linear Functions

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