Video: Transformations of Graphs

The red graph in the figure represents the equation 𝑦 = 𝑓(𝑥) and the blue graph represents the equation 𝑦 = 𝑔(𝑥). Express 𝑔(𝑥) as a transformation of 𝑓(𝑥).

02:20

Video Transcript

The red graph in the figure represents the equation 𝑦 equals 𝑓 of 𝑥, and the blue graph represents the equation 𝑦 equals 𝑔 of 𝑥. Express 𝑔 of 𝑥 as a transformation of 𝑓 of 𝑥.

Our red function is our 𝑓 of 𝑥, and the function in blue is 𝑔 of 𝑥. For this transformation, we start with the red function 𝑓 of 𝑥. And some transformation produces the blue line, the 𝑔 of 𝑥 function.

When we look at the two lines in this image, we see that the blue line is skinnier than the red line. Mathematically, we say that is compressed. Both of these functions show the 𝑥-intercept at point zero, zero. The red function has another intercept at negative two, zero. And our blue function has an intersect at negative one, zero.

The red function here is wider than the blue function. The same thing is true on the positive 𝑥-axis: 𝑓 of 𝑥 has an intercept at one, zero and 𝑔 of 𝑥 has an intercept at one-half, zero. 𝑓 of 𝑥 has been compressed into 𝑔 of 𝑥. 𝑓 of 𝑥 has been compressed by a factor of one-half in the 𝑥-direction.

We also have to make sure that we know that the width is the only thing being compressed. This function is not being compressed in the direction of the 𝑦-axis. They are not any taller. And that means 𝑔 of 𝑥 is created.

Taking the 𝑓 of 𝑥 and multiplying each 𝑥-value by two creates this compression. If we want to compress a function by one over two, then we’ll need to multiply each 𝑥-value by two. 𝑔 of 𝑥 equals 𝑓 of two 𝑥.

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