Dilations

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 bug-eyed 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 x-axis while the purple flashing buttons represent how a function can dilate on the y-axis.

How do you dilate the function y = f(x) ?

DilationAnimationAll

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 y-values) by a or you can multiply the inside of the function (the x-values) 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:

  1. Y, a+

  2. Y, a-

  3. X, a+

  4. 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 y-stretch

Example: y = 4|x| will stretch the y-values by a multiple of 4

(Note: Like Translations, the dilation effect on the y-values is exactly as you see it… multiplying by 4 multiplies the y-values by 4.)

 DilationYa+

If you press the Y button and () button, the a value becomes smaller than 1 and the result is a y-compression

Example: y = 1/4|x| will compress the y-values by a multiple of 1/4

(Note: Like Translations, the dilation effect on the y-values is exactly as you see it… multiplying by 1/4 multiplies the y-values by 1/4.)

DilationYa-

X AXIS DILATION

If you press the X button and (+) button, the a value becomes greater than 1 and the result is an x-compression

Example: y = |4x| will compress the original function by a value of 4 on the x-axis

(Note: Like Translations, the dilation effect on the x-values is opposite.)

DilationXa+

If you press the X button and () button, the a value becomes smaller than 1 and the result is an x-stretch

Example: y = |1/4x| will stretch the original function by 1/4 on the x-axis

(Note: Like Translations, the dilation effect on the x-values is opposite.)

DilationXa-

 

 

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:

  1. A y-stretch can look like an x-compression (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 ‘+’.
  2. y-compress can look like an x-stretch (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 ‘-‘.

Now let’s learn about Reflections