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 𝑥 minus one plus four.
We need to recall firstly what types of transformations are represented by 𝑓 of 𝑥 minus one plus four. Firstly, we recall that 𝑓 of 𝑥 plus 𝑏 is a translation of 𝑓 of 𝑥 𝑏 units in the positive 𝑦-direction. So, when we’re adding a constant to the overall function, this is a vertical translation by that amount. It simply shifts the graph up or down by the value of the constant 𝑏. So, adding four on the end of the function here means that we are translating 𝑦 equals 𝑓 of 𝑥 four units up.
But we’ve also got something happening inside the parentheses to the input of the function because we have 𝑓 of 𝑥 minus one. We should recall then that when a change is happening to the input to the function itself, this has a horizontal effect. 𝑓 of 𝑥 minus 𝑎 is a translation 𝑎 units in the positive 𝑥-direction. It’s a horizontal shift this time. So, 𝑓 of 𝑥 minus one means that the graph is being translated one unit in the positive 𝑥-direction. That’s one unit to the right. So, we could say that this transformation is a translation one unit right and four units up, or we could say that we’re translating by the vector one, four.
Either way, we need to consider what the impact is specifically on this point 𝐴 which has coordinates two, one. If we translate this point one unit right and then four units up, we arrive at a new point here which we can think of as 𝐴 prime. The coordinates of this point are three, five. To achieve the horizontal translation, we’ve added one to the 𝑥-coordinate of two. And to achieve the vertical translation, we’ve added four to the 𝑦-coordinate of one. So, we find that the corresponding local maximum for the transformation 𝑦 equals 𝑓 of 𝑥 minus one plus four is the point three, five.