Question Video: Finding the Coordinates of a Point on a Graph Following a Dilation Mathematics

Video Transcript

The figure shows the graph of 𝑦 equals 𝑓 of 𝑥 and the point 𝐴. The point 𝐴 is a local maximum. Identify the corresponding local maximum for the transformation 𝑦 equals 𝑓 of two 𝑥.

Let’s begin by recalling the transformation that maps the graph of 𝑦 equals 𝑓 of 𝑥 onto the graph of 𝑦 equals 𝑓 of two 𝑥. We know that for a function 𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑓 of 𝑏 times 𝑥 is a horizontal dilation by a scale factor of one over 𝑏. And so, if we compare our equation, that’s 𝑦 equals 𝑓 of two 𝑥, to the general equation, 𝑦 equals 𝑓 of 𝑏𝑥, we can see that we’re going to have a horizontal dilation. Let’s work the scale factor out by letting 𝑏 be equal to two.

When we do, we see that for our function 𝑦 equals 𝑓 of 𝑥, 𝑦 equals 𝑓 of two 𝑥 is a horizontal dilation by a scale factor of one-half. In other words, we’re going to compress our graph about the 𝑦-axis. When we do, it looks a little something like this. We’ll call the coordinate of our local maximum on our transformation 𝐴 prime as shown. We can see that its 𝑦-coordinate remains unchanged, but its 𝑥-coordinate is halved. So, 𝐴 prime is two over two, one or one, one. And so, we see that the corresponding local maximum for the transformation 𝑦 equals 𝑓 of two 𝑥 is one, one.