Vertical Transformations
How does adding to , or multiplying by , transform the graph of ?
Set the large vertical slider to the bottom, "Step 1". Set the small horizontal selector switch at the top of the right-hand pane to line up with either the slider or the slider, The slider will create vertical translations (shifts), while the slider controls vertical 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. In Step 3, we find the -value of the original function by plugging into . Next, in Step 4, we modify the -value by adding or multiplying it by , depending on how the selector switch is set. Thus, we have or as a new -value. In Step 5, we plot the new -value back at 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?