Here are solutions to the questions on the worksheet accompanying the section Post-Composition and Graphing.
| Corresponding Algebraic formula |
x | |x| | 2|x| | -2|x| | -2|x| + 1 |
|---|---|---|---|---|---|
| Numerical Effect | Take the input | Apply the absolute value function |
Multiply by 2 | Multiply by -1 | Add 1 |
| Numerical Results |
Inputs | Outputs | |||
| -1 | |||||
| 0 | |||||
| 1 | |||||
so that f goes through the points:
| x | y = f(x) = -2|x| + 1 |
|
-1 0 1 |
| Corresponding Algebraic formula |
x | |x| | 2|x| | -2|x| | -2|x| + 1 |
|---|---|---|---|---|---|
| Numerical Effect | Take the input | Apply the absolute value function |
Multiply by 2 | Multiply by -1 | Add 1 |
| Numerical Results |
Inputs | Outputs | |||
| -1 | 1 | 2 | -2 | -1 | |
| 0 | 0 | 0 | 0 | 1 | |
| 1 | 1 | 2 | -2 | -1 | |
Thus, the graph of f goes through the points:
| x | y = f(x) = -2|x| + 1 |
|
-1 0 1 |
-1 1 -1 |
| Corresponding Algebraic formula |
y = |x| | y = 2|x| | y = -2|x| | y = -2|x| + 1 |
|---|---|---|---|---|
| Geometric Effect | Take the graph
of the absolute value function |
Stretch vertically by a factor of 2 |
| Corresponding Algebraic formula |
y = |x| | y = 2|x| | y = -2|x| | y = -2|x| + 1 |
|---|---|---|---|---|
| Geometric Effect | Take the graph
of the absolute value function |
Stretch vertically by a factor of 2 |
Flip the vertical axis | Shift vertically, up by 1 |
- Notice that the final graph does go through the points we calculated in the previous Exercise.
| Corresponding Algebraic formula |
x | |x| | |||
|---|---|---|---|---|---|
| Geometric Effect | Take the graph
of the absolute value function |
||||
| Numerical Effect | Take the input | Apply the absolute value function |
|||
| Numerical Results |
Inputs | Outputs | |||
| -1 | |||||
| 0 | |||||
| 1 | |||||
| Corresponding Algebraic formula |
x | ||||
|---|---|---|---|---|---|
| Geometric Effect | Take the graph
of the square root function |
||||
| Numerical Effect | Take the input | Apply the square root function |
|||
| Numerical Results |
Inputs | Outputs | |||
| 0 | |||||
| 1 | |||||
| 4 | |||||
so that h goes through the points:
| x | |
|
0 1 4 |
Plot the points you calculated for y = h(x) and use the the geometric description to envision how to correctly connect-the-dots to create a fairly accurate sketch of the resulting graph. Hint: Imagine, in your mind's eye what the arithmetic operations would do geometrically to the graph of the square root function; if necessary, make a rough sketch of the graph after each operation.
| Corresponding Algebraic formula |
x | ||||
|---|---|---|---|---|---|
| Geometric Effect | Take the graph
of the square root function |
Stretch vertically by a factor of 2 |
Flip the vertical axis |
Shift vertically, up by 1 |
|
| Numerical Effect | Take the input | Apply the square root function |
Multiply by 2 | Multiply by -1 | Add 1 |
| Numerical Results |
Inputs | Outputs | |||
| 0 | 0 | 0 | 0 | 1 | |
| 1 | 1 | 2 | -2 | -1 | |
| 4 | 2 | 4 | -4 | -3 | |
so that h goes through the points:
| x | |
|
0 1 4 |
1 -1 -3 |
As before, stretching the graph of the square root function vertically by 2 will make the graph steeper, leaving the vertex at the origin. Flipping the vertical axis will keep the graph as steep, but the graph will now go downwards from the same vertex. Finally, shifting up by 1 will simply move the graph up 1 unit; as before, this is most clearly visible at the vertex. As before, by selecting the series of Examples from the following applet, we can verify this analysis:
| Corresponding Algebraic formula |
x | ||||
|---|---|---|---|---|---|
| Geometric Effect | Take the graph
of the square root function |
||||
| Numerical Effect | Take the input | Apply the square root function |
|||
| Numerical Results |
Inputs | Outputs | |||
| 0 | |||||
| 1 | |||||
| 4 | |||||
| Corresponding Algebraic formula |
x | ëxû | 2ëxû | -2ëxû | -2ëxû + 1 |
|---|---|---|---|---|---|
| Geometric Effect | Take the graph
of the greatest integer function |
||||
| Numerical Effect | Take the input | Apply the greatest integer function |
|||
| Numerical Results |
Inputs | Outputs | |||
| -1 | |||||
| 0 | |||||
| 1 | |||||
so that r goes through the points:
| x | y = r(x) = -2ëxû + 1 |
|
-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 |
x | ëxû | 2ëxû | -2ëxû | -2ëxû + 1 |
|---|---|---|---|---|---|
| Geometric Effect | Take the graph
of the greatest integer function |
Stretch vertically by a factor of 2 |
Flip the vertical axis |
Shift vertically, up by 1 |
|
| Numerical Effect | Take the input | Apply the greatest integer function |
Multiply by 2 | Multiply by -1 | Add 1 |
| Numerical Results |
Inputs | Outputs | |||
| -1 | -1 | -2 | 2 | 3 | |
| 0 | 0 | 0 | 0 | 1 | |
| 1 | 1 | 2 | -2 | -1 | |
so that r goes through the points:
| x | y = r(x) = -2ëxû + 1 |
|
-1 0 1 |
3 1 -1 |
This time, stretching the graph of the greatest integer function vertically by 2 will pull the "steps" farther apart (i.e., by 2 units). When we flip the vertical axis, the "steps" will now go downwards. Finally, shifting up by 1 will simply move each step up 1 unit. Unlike the previous examples, this graph does not have a "vertex". Although we cannot simply "connect-the-dots", if we have a rough picture of the graph, we can sketch the horizontal lines from (-1, 3), (0, 1), and (1, -1), and then extend this pattern to obtain the rest of the graph. Remember to put the closed circles on the left side of each step, with an open circle on the right. As usual, we can verify this analysis by examining the Examples in the following applet:
| Corresponding Algebraic formula |
x | ëxû | 3ëxû | 3ëxû - 5 |
|---|---|---|---|---|
| Geometric Effect | Take the graph
of the greatest integer function |
Stretch vertically by a factor of 3 |
Shift down 5 units |
|
| Numerical Effect | Take the input | Apply the greatest integer function |
Multiply by 3 | Subtract 5 |
| Numerical Results |
Inputs | Outputs | ||
| -1 | -1 | -3 | -8 | |
| 0 | 0 | 0 | -5 | |
| 1 | 1 | 3 | -2 | |
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 another 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 Graphing and Linear Functions
| Table of Contents | Glossary |