If you’ve ever had an optometrist dilate your eyes, you know that your pupils turn large and buggy. Creepy, right!?
Much like getting your eyes dilated, parent functions become bugeyed too; that is… magnified in size! Did you know there are four main ways in which a parent function dilates? To help illustrate this process, think of a game controller like the one shown below. The green flashing buttons represent how a function can dilate on the xaxis while the purple flashing buttons represent how a function can dilate on the yaxis.
How do you dilate the function y = f(x) ?
Multiply ‘a’ to y = f(x) to become
y = af(x) or y = f(ax)
As you can see, you can multiply the function (the yvalues) by a or you can multiply the inside of the function (the xvalues) by a. Not only that, but you can choose the positive value of a by pressing the ‘+‘ or ‘–‘ buttons. If you press the ‘+‘ button, you make a bigger than 1 (like 2 or 450.9 or anything). If you press the ‘–‘ button, you make a smaller than 1 (like 0.25 or 0.50 or anything). As a result, pushing these flashing buttons create 4 combinations:

Y, a+

Y, a

X, a+

X, a
Y AXIS DILATION
If you press the Y button and (+) button, the a value becomes greater than 1 and the result is a ystretch
Example: y = 4x will stretch the yvalues by a multiple of 4
(Note: Like Translations, the dilation effect on the yvalues is exactly as you see it… multiplying by 4 multiplies the yvalues by 4.)
If you press the Y button and (–) button, the a value becomes smaller than 1 and the result is a ycompression
Example: y = 1/4x will compress the yvalues by a multiple of 1/4
(Note: Like Translations, the dilation effect on the yvalues is exactly as you see it… multiplying by 1/4 multiplies the yvalues by 1/4.)
X AXIS DILATION
If you press the X button and (+) button, the a value becomes greater than 1 and the result is an xcompression
Example: y = 4x will compress the original function by a value of 4 on the xaxis
(Note: Like Translations, the dilation effect on the xvalues is opposite.)
If you press the X button and (–) button, the a value becomes smaller than 1 and the result is an xstretch
Example: y = 1/4x will stretch the original function by 1/4 on the xaxis
(Note: Like Translations, the dilation effect on the xvalues is opposite.)
Making the Connection
Hopefully you notice that there are four dilation combinations; however, there are two main ways a parent function can appear… tall or wide:
 A ystretch can look like an xcompression (and vice versa). The functions appear taller than the original parent function. You should have also noticed that both of these dilations occur when ‘a‘ is increasing ‘+’.
 A ycompress can look like an xstretch (and vice versa). The functions appear wider than the original parent function. You should have also noticed that both of these dilations occur when ‘a‘ is decreasing ‘‘.