Horizontal Transformations

Why does modifying the value of a function transform its graph? Why does make shift left while shifts it right? Why does ( compress while stretches ? This seems backwards, doesn't it?
Set the large vertical slider to the bottom, "Step 1". Set the small vertical selector switch at the top of the right-hand pane to line up with either the slider or the slider, The slider will create horizontal translations (shifts), while the slider controls horizontal dilations (stretches/compressions). In Step 1, we see our original function in purple, called . (Feel free to enter a different function in the "f(x) =" box). Move the Step slider up one notch. In Step 2, we can select any value of in 's domain by dragging the red point. Next, in Step 3, we modify the -value by subtracting or multiplying it by , depending on how the selector switch is set. Notice that if we subtract a negative value of , we are actually adding a positive number to , and thus we are moving to the right. Thus, we have or as a new -value. In Step 4 we take this new -value and plug it into , giving us the -value that goes with or . But remember that we want the transformation or : in other words, we want the -value at to be the same as the -value at or . So in Step 5, we "pull" the new -value back to the original -value. In Step 6, we see in blue what we get when we do these steps for all values of . (The original is shown dashed). Once you've understood the six steps in this app, leave the slider on Step 6 and move the other sliders to see these transformations in action. Another exercise you can do is to return the slider to Step 1, enter a new function, and then predict what will happen at each successive step, checking your prediction as you go. Above the "Step" slider, you'll see a brief description of what is happening at that step. Special cases to think about: What happens when and when ? Why?