Here are solutions to the questions on the worksheet accompanying the section Pre-Composition and Graphing.
Complete the following tables to calculate the three points on the graph of f, which correspond to these same three points; that is, calculate numerical values of the three inputs that yield the same outputs. Also, record the geometric effect of each step. Remember: Since we view each change as affecting the inputs of the function, all geometric effects will be in the horizontal direction.
| Corresponding Algebraic formula |
|||||
|---|---|---|---|---|---|
| Geometric Effect | Original | Shift left 1 | |||
| Numerical Effect | Subtract 1 | Divide by -1 | Divide by 2 | ||
| Numerical Results |
Inputs | Outputs | |||
| 0 | 0 | ||||
| 1 | 1 | ||||
| 4 | 2 | ||||
so that f goes through the points:
| x | |
| 0 1 2 |
Note: In the process, you are actually determining the graphs of each intermediate expression.
| Corresponding Algebraic formula |
|||||
|---|---|---|---|---|---|
| Geometric Effect | Original | Shift left 1 | Flip horizontally | Shrink by a factor of 2 horizontally |
|
| Numerical Effect | Subtract 1 | Divide by -1 | Divide by 2 | ||
| Numerical Results |
Inputs | Outputs | |||
| 0 | -1 | 1 | 1/2 | 0 | |
| 1 | 0 | 0 | 0 | 1 | |
| 4 | 3 | -3 | -3/2 | 2 | |
Thus, the graph of f goes through the points:
| x | |
|
1/2 0 -3/2 |
0 1 2 |
| Corresponding Algebraic formula | |||||
|---|---|---|---|---|---|
| Geometric Effect | Original | ||||
| Numerical Effect | |||||
| Numerical Results |
Inputs | Outputs | |||
| 0 | 0 | ||||
| 1 | 1 | ||||
| 4 | 2 | ||||
| Corresponding Algebraic formula |
|x| | |x + 1| | |-x + 1| | |-2x + 1| | |
|---|---|---|---|---|---|
| Geometric Effect | Original | ||||
| Numerical Effect | |||||
| Numerical Results |
Inputs | Outputs | |||
| -1 | 1 | ||||
| 0 | 0 | ||||
| 1 | 1 | ||||
so that h goes through the points:
| x | y = h(x) = |-2x + 1| |
| 1 0 1 |
Plot the points you calculated for y = h(x) and use your geometric description to envision how to correctly connect-the-dots to create a fairly accurate sketch of the resulting graph.
| Corresponding Algebraic formula |
|x| | |x + 1| | |-x + 1| | |-2x + 1| | |
|---|---|---|---|---|---|
| Geometric Effect | Original | Shift left 1 | Flip horizontally | Shrink horizontally by a factor of 2 |
|
| Numerical Effect | Subtract 1 | Divide by -1 | Divide by 2 | ||
| Numerical Results |
Inputs | Outputs | |||
| -1 | -2 | 2 | 1 | 1 | |
| 0 | -1 | 1 | 1/2 | 0 | |
| 1 | 0 | 0 | 0 | 1 | |
so that h goes through the points:
| x | y = h(x) = |-2x + 1| |
|
1 1/2 0 |
1 0 1 |
As before, shifting the graph of the absolute value function to the left by 1 will shift the vertex to (-1, 0), but it will still rise at a 45° angle to both sides. After we flip this horizontally, the vertex will flip to (1, 0). Finally, when we shrink horizontally (towards the origin), the graph will become "compressed"; this will "shrink" the vertex to (1/2, 0) and the graph will get "narrower". Note: It is no accident that this could equally well be described as "steeper"; we will explore this further in the next section. We can verify this analysis by selecting the Examples from the following applet:
| Corresponding Algebraic formula |
|x| | |x - 2| | |(1/3)x - 2| | |
|---|---|---|---|---|
| Geometric Effect | Original | Shift right 2 units | Stretch horizontally by a factor of 3 |
|
| Numerical Effect | Add 2 | Multiply by 3 | ||
| Numerical Results |
Inputs | Outputs | ||
| -1 | 1 | 3 | 1 | |
| 0 | 2 | 6 | 0 | |
| 1 | 3 | 9 | 1 | |
You should use the Multigraph Utility to verify that at each step goes through the points you have calculated and that the graph has the stated geometric relationship to the graph at the previous step. In this example, we would plot (see the Examples in the top menu):
| Corresponding Algebraic formula |
ëxû | ëx + 1û | ë-x + 1û | ë-2x + 1û | |
|---|---|---|---|---|---|
| Geometric Effect | Original | ||||
| Numerical Effect | |||||
| Numerical Results |
Inputs | Outputs | |||
| -1 | -1 | ||||
| 0 | 0 | ||||
| 1 | 1 | ||||
so that r goes through the points:
| x | y = r(x) = ë-2x + 1û |
|
1 1/2 0 |
-1 0 1 |
Plot the points you calculated for y = r(x) and use the the geometric description to envision how to draw in the rest of the resulting graph. In particular, you should be careful to correctly place the open and closed circles on the graph.
| Corresponding Algebraic formula |
ëxû | ëx + 1û | ë-x + 1û | ë-2x + 1û | |
|---|---|---|---|---|---|
| Geometric Effect | Original | Shift left 1 | Flip horizontally | Shrink horizontally by a factor of 2 |
|
| Numerical Effect | Subtract 1 | Divide by -1 | Divide by 2 | ||
| Numerical Results |
Inputs | Outputs | |||
| -1 | -2 | 2 | 1 | -1 | |
| 0 | -1 | 1 | 1/2 | 0 | |
| 1 | 0 | 0 | 0 | 1 | |
so that r goes through the points:
| x | y = r(x) = ë-2x + 1û |
|
1 1/2 0 |
-1 0 1 |
Shifting left by 1 is easy to visualize; each "step" moves left 1 unit. When we flip horizontally, the steps will now go down as we move to the right. Moreover, the closed circles will now appear on the right side of each step. Finally, shrinking the graph horizontally by 2 will shrink each "step" to 1/2 a unit long, and bring them closer to the vertical axis; this will make the graph look somewhat "steeper". Again, we cannot simply "connect-the-dots", but, since we have a rough picture of the graph, we can sketch the horizontal lines from (1, -1), (1/2, 0), and (0, 1), and then extend this pattern to obtain the rest of the graph. Remember to put the closed circles on the right side of each step, with an open circle on the left. As usual, we can verify this analysis by selecting the Example from the following applet:
| Corresponding Algebraic formula |
|x| | ëx - 1û | ë-x - 1û | ë(-1/2)x - 1û | |
|---|---|---|---|---|---|
| Geometric Effect | Original | Shift right 1 unit | Flip horizontally | Stretch horizontally by a factor of 2 |
|
| Numerical Effect | Add 1 | Negate | Multiply by 2 | ||
| Numerical Results |
Inputs | Outputs | |||
| -1 | 0 | 0 | 0 | -1 | |
| 0 | 1 | -1 | -2 | 0 | |
| 1 | 2 | -2 | -4 | 1 | |
You should use the Multigraph Utility to verify that at each step goes through the points you have calculated and that the graph has the stated geometric relationship to the graph at the previous step. In this example, we would plot (see the Examples in the top menu):
Note: You should repeat this Exercise, even experimenting with different "core" functions, as often as necessary, until you feel confident that you have mastered this graphing technique; if you use an other function (such as the cube root or x2), you will need to decide what three points would be important to plot by examining its graph.
Go to Transformational Graphing
| Table of Contents | Glossary |