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.
