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What do you notice? What do you wonder?
Here is the graph of function and a table of values.
| -3 | 0 | |
| -1.5 | -4.3 | |
| -1 | -4 | |
| 0 | -1.8 | |
| 0.6 | 0 | |
| 2.6 | 3.9 | |
| 4 | 0 |
Here is another copy of the graph of from the earlier activity. This time, let be the function defined by .
Complete the tables. The values for will not be the same for the two tables.
| -3 | 0 |
| -1.5 | -4.3 |
| -1 | -4 |
| 0 | -1.8 |
| 0.6 | 0 |
| 2.6 | 3.9 |
| 4 | 0 |
Here are graphs of the functions , , and , where and . How do these equations match the transformation we see from to and from to ?
Considering first the equation , we know that for the same input , the value of will be the opposite of the value of . For example, since , we know that . We can see this relationship in the graphs where is the reflection of across the -axis.
Looking at , this equation tells us that the two functions have the same output for opposite inputs. For example, 1 and -1 are opposites, so (and is also true!). We can see this relationship in the graphs where is the reflection of across the -axis.