Stretching & Compressing a Function

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  • 0:00 Stretching and…
  • 0:39 Vertical Stretching &…
  • 3:07 Horizontal Stretching…
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Lesson Transcript
Instructor: Elizabeth Foster

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

Learn about the changes to a function that can ether stretch or compress the graph of that function. This lesson covers horizontal and vertical changes, including both stretching and compression.

Stretching and Compressing Functions

You knew you could graph functions. The x-values, or input, of the function go on the x-axis of the graph, and the f(x) values also called y-values, or output, go on the y-axis of the graph. But did you know that you could stretch and compress those graphs, vertically and horizontally?

Simple changes to the equation of a function can change the graph of the function in predictable ways. In this lesson, we'll go over four different changes: vertical stretching, vertical compression, horizontal stretching, and horizontal compression.

Vertical Stretching & Compression

Instead of starting off with a bunch of math, let's start thinking about vertical stretching and compression just by looking at the graphs. Vertical stretching means the function is stretched out vertically, so it's taller. Vertical compression means the function is squished down vertically, so it's shorter. That's what stretching and compression actually look like. Now it's time to get into the math of how we can change the function to stretch or compress the graph.

On the graph of a function, the F(x), or output values of the function, are plotted on the y-axis. A function that is vertically stretched has bigger y-values for any given value of x, and a function that is vertically compressed has smaller y-values for any given value of x.

For example, look at the graph of a stretched and compressed function.

Vertical stretch and compression

Look at the value of the function where x = 0. You can see that for the original function where x = 0, there's some value of y that's greater than 0. For the stretched function, the y-value at x = 0 is bigger than it is for the original function. For the compressed function, the y-value is smaller.

Now let's look at what kinds of changes to the equation of the function map onto those changes in the graph.

To vertically stretch a function, multiply the entire function by some number greater than 1. This is basically saying that whatever you would ordinarily get out of the function as a y-value, take that and multiply it by 2 or 3 or 4 to get the new, higher y-value. This is how you get a higher y-value for any given value of x.

To vertically compress a function, multiply the entire function by some number less than 1. This is the opposite of vertical stretching: whatever you would ordinarily get out of the function, you multiply it by 1/2 or 1/3 or 1/4 to get the new, smaller y-value.

That's great, but how do you know how much you're stretching or compressing the function? In general, if y = F(x) is the original function, then you can vertically stretch or compress that function by multiplying it by some number a:

If a > 1, then aF(x) is stretched vertically by a factor of a. For example, if you multiply the function by 2, then each new y-value is twice as high.

If 0 < a < 1, then aF(x) is compressed vertically by a factor of a.

Horizontal Stretching & Compression

That was how to make a function taller and shorter. But what about making it wider and narrower? That's horizontal stretching and compression. Let's look at horizontal stretching and compression the same way, starting with the pictures and then moving on to the actual math.

Horizontal stretch and compression

Horizontal stretching means that you need a greater x-value to get any given y-value as an output of the function. You can see this on the graph. See how the maximum y-value is the same for all the functions, but for the stretched function, the corresponding x-value is bigger.

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