How does adding [math]k[/math] to [math]f(x)[/math], or multiplying [math]a[/math] by [math]f(x)[/math], transform the graph of [math]f(x)[/math]?

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 [math]a[/math] slider or the [math]k[/math] slider, The [math]k[/math] slider will create vertical [i]translations [/i](shifts), while the [math]a[/math] slider controls vertical [i]dilations[/i] (stretches/compressions). [br][br]In Step 1, we see our original function in purple, called [math]f(x)[/math]. (Feel free to enter a different function in the "[color=#000000]f(x) =[/color]" box). Move the Step slider up one notch. In Step 2, we can select any value of [math]x[/math] in [math]f[/math]'s domain by dragging the red [color=#ff0000][math]x[/math][/color] point. In Step 3, we find the [math]y[/math]-value of the original function by plugging [math]x[/math] into [math]f[/math]. Next, in Step 4, we modify the [math]y[/math]-value by adding [math]k[/math] or multiplying it by [math]a[/math], depending on how the selector switch is set. Thus, we have [math]f(x)+k[/math] or [math]a \times f(x)[/math] as a new [math]y[/math]-value. In Step 5, we plot the new [math]y[/math]-value back at the original [math]x[/math]-value. In Step 6, we see in blue what we get when we do these steps for all values of [math]x[/math]. (The original [math]f(x)[/math] is shown dashed).[br][br]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.[br][br]Special cases to think about: What happens when [math]a=0[/math] and when [math]k=0[/math]? Why?