Dilations are transformations that squish or stretch curves. On this page, we'll learn how to squish and stretch the sine function vertically and horizontally.

#### First,

Enter y = n sin x.
Change the slider values to match the illustration below.

Your job is to figure out what n does as fully as possible. Move the slider. Play the animation. At first we set it up so that you look only at positive values of n. But you can change that.

#### Here are some questions you can explore:

• What happens to the curve as n gets bigger?
• What happens to the curve if n is negative?

#### Now let's look at a different version of the sine function.

Enter y =sin(nx).
Do exactly the same exploration. When you understand the sine function, try a different function, such as .

1. What's the rule for stretching a function horizontally?
2. What's the rule for squishing a function horizontally?
3. What's the rule for stretching a function vertically?
4. What's the rule for squishing a function vertically?

 One lesson should be the following: To affect a curve horizontally, you have to act upon every x in the function; to affect it vertically, you have to act upon the whole expression (which amounts to doing something to every y, if you want to look at it that way). Another observation: Addition causes translation; multiplication causes dilation.

© 1996 Key Curriculum Press