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The table shows Clare’s elevation on a Ferris wheel at different times, . Clare got on the ride 80 seconds ago. Right now, at time 0 seconds, she is at the top of the ride. Assuming the Ferris wheel moves at a constant speed for the next 80 seconds, complete the table.
| time (seconds) | height (feet) |
|---|---|
| -80 | 0 |
| -60 | 31 |
| -40 | 106 |
| -20 | 181 |
| 0 | 212 |
| 20 | |
| 40 | |
| 60 | |
| 80 |
We've learned how to transform functions in several ways. We can translate graphs of functions up and down, changing the output values while keeping the input values. We can translate graphs left and right, changing the input values while keeping the output values. We can reflect functions across an axis, swapping either input or output values for their opposites depending on which axis is reflected across.
For some functions, we can perform specific transformations and it looks like we didn't do anything at all. Consider the function whose graph is shown here:
What transformation could we do to the graph of that would result in the same graph? Examining the shape of the graph, we can see a symmetry between points to the left of the -axis and the points to the right of the -axis. Looking at the points on the graph where and , these opposite inputs have the same outputs since and . This means that if we reflect the graph across the -axis, it will look no different. This type of symmetry means is an even function.
Now consider the function whose graph is shown here:
Graph of function g. X axis from negative 2 to 2, by 1's. Y axis from negative 8 to 6, by 2’s. From left to right, the function begins in the third quadrant, moves upward, passing through negative 1 comma 2 point 3 5, and through 0 comma 0. It continues upward and to the right passing through 1 comma 2 point 3 5 and continues to move upward and to the right, ending in the first quadrant.
What transformation could we do to the graph of that would result in the same graph? Examining the shape of the graph, we can see that there is a symmetry between points on opposite sides of the axes. Looking at the points on the graph where and , these opposite inputs have opposite outputs since and . So a transformation that takes the graph of to itself has to reflect across the -axis and the -axis. This type of symmetry is what makes an odd function.
A function that satisfies the condition for all inputs . You can tell an even function from its graph: Its graph is symmetric about the -axis.
A function that satisfies for all inputs . You can tell an odd function from its graph: Its graph is taken to itself when you reflect it across both the - and -axes. This can also be seen as a 180 rotation about the origin.