Question Video: Identifying the Image of a Point on a Trigonometric Graph following a Transformation Mathematics

Video Transcript

The figure shows the graph of 𝑓 of 𝑥. A transformation maps 𝑓 of 𝑥 to two 𝑓 of 𝑥. Determine the coordinates of 𝐴 following this transformation.

And then we have a graph, which gives us coordinates of 𝐴 as 90, one. So to answer this question, let’s begin by thinking about the algebraic representation of transformations. Specifically, which transformation maps 𝑓 of 𝑥 to two 𝑓 of 𝑥? Well, suppose we have the function 𝑓 of 𝑥. This is mapped onto 𝑎𝑓 of 𝑥 by a vertical stretch, scale factor 𝑎. This means we can map 𝑓 of 𝑥 onto two 𝑓 of 𝑥 by a vertical stretch with a scale factor of two. And whilst the axes don’t quite go up far enough, we can sketch this approximately on our diagram. A vertical stretch by a scale factor of two will maintain the same 𝑥-intercepts.

For all other coordinates, the 𝑦-values will essentially be doubled. And so if we were to draw the graph of 𝑦 equals two 𝑓 of 𝑥, it might look a little something like this. Let’s then say that point 𝐴 is mapped onto point 𝐴 prime following this transformation. What are the coordinates of point 𝐴 prime? Well, we said actually the 𝑥-coordinates stay the same, and what happens is the 𝑦-coordinates are all multiplied by two. So 𝐴 prime must have coordinates 90, one times two, which is 90, two.

Now, in fact, if we think about the equation of each graph, this makes a lot of sense. Notice that our original graph is sinusoidal. It has maxima and minima of one and negative one, respectively, and a 𝑦-intercept of zero. It also appears to repeat; it’s periodic with a period of 360 degrees. In fact, the function 𝑓 of 𝑥 is sin of 𝑥. This means that two 𝑓 of 𝑥 must simply be two sin of 𝑥.

So after the transformation, we have to plot the graph 𝑦 equals two sin of 𝑥. We might do that using a table or any other suitable method. Either way, we should observe that point 𝐴 maps onto point 𝐴 prime with coordinates 90, two. The coordinates of 𝐴 following our transformation then are 90, two.