Video Transcript
The figure shows the graph of 𝑦 equals 𝑓 of 𝑥 and point 𝐴, which is a local maximum. Identify the corresponding local maximum for the transformation 𝑦 equals 𝑓 of 𝑥 minus three.
We need to recall firstly our rules for transformations and what 𝑓 of 𝑥 minus three represents. We should recall that 𝑓 of 𝑥 minus 𝑎 for some constant 𝑎 is a translation 𝑎 units in the positive 𝑥-direction. The graph of 𝑓 of 𝑥 minus 𝑎 will be the graph of 𝑓 of 𝑥 but simply shifted or moved 𝑎 units to the right. In this specific example, we’ve been given the value of 𝑎 is three. So we’re looking at a translation three units in the positive 𝑥-direction. We’re also asked specifically to identify where the point 𝐴, which is a local maximum on the original graph, is translated to.
Well, if we are translating the graph three units to the right, then the 𝑥-coordinate will increase by three, but the 𝑦-coordinate will be unaffected. The new 𝑥-value will therefore be two plus three, and the new 𝑦-value will be the same as it was before. The point two, one is therefore translated to become the point five, one. So we found that if 𝐴 is the local maximum for the graph 𝑦 equals 𝑓 of 𝑥, then the corresponding local maximum for the transformation 𝑦 equals 𝑓 of 𝑥 minus three is the point five, one.
