Answer: y = f(x - 4) - 1, lower left corner
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Explanation:
Let's focus on only the left-most points of each curve
For the black semicircle, the left-most point is located at (-7,0). The corresponding point on the red semicircle is at (-3,-1)
To go from (-7,0) to (-3,-1), we will do two things
- Shift 4 units to the right
- Shift 1 unit down
This can be done in any order. The action of "shift 4 units to the right" is effectively the same as shifting the xy axis 4 units to the left while keeping the curves fixed in place. This gives the illusion of movement. When we shift 4 the xy axis 4 units to the left, we're replacing each x input with x-4 as the new input.
In short, we go from f(x) to f(x-4). This is why we have this opposite motion going on.
The up and down motion is fairly straight forward. To shift 1 unit down, we subtract off 1. This is to subtract 1 from the y coordinate.
Overall, we go from y = f(x) to y = f(x - 4) - 1
Applying this transformation will move (-7,0) to (-3,-1) as described above. It will also move every other point on the black curve to their new corresponding location on the red curve.
