Video Transcript
The figure shows the graph of a function. Which of the following equations represents the graph? Is it (A) 𝑦 equals two sin 𝑥? (B) 𝑦 equals sin of 𝑥 plus two. (C) 𝑦 equals sin two 𝑥. (D) 𝑦 equals sin 𝑥 minus two. Or (E) 𝑦 is equal to sin of 𝑥 minus two.
Let’s begin by having a look at all of the possible options we have here. Each of these options is based on the function sin of 𝑥. So let’s begin by plotting the graph of 𝑦 equals sin of 𝑥 and see if we can work out what sort of transformation will obtain the new graph. First, we know that the graph of 𝑦 equals sin of 𝑥 has minima and maxima at negative one and one, respectively. It has a period of 360 degrees or two 𝜋 radians. And it passes through the 𝑦-axis at zero. So one full period of the graph of 𝑦 equals sin 𝑥 looks like this.
Let’s continue this graph as shown. So in order to identify the graph that we were given, let’s begin by identifying what transformation maps 𝑦 equals sin of 𝑥 onto this graph. Now, if we look at the relative maxima and minima, we see that the graph appears to have been shifted right by some number of units. In fact, if we take the point zero, zero, we see this is mapped onto the point two, zero on the new graph. In other words, it appears as if a horizontal translation two units to the right has occurred. Let’s describe this using vector notation as the vector two, zero.
So let’s think about the algebraic representation of such transformation. We know that the function 𝑓 of 𝑥 maps onto the function 𝑓 of 𝑥 minus 𝑎 by a horizontal translation by 𝑎 units to the right, in other words by the vector 𝑎, zero. This means if we have the function 𝑓 of 𝑥, this can be mapped onto 𝑓 of 𝑥 minus two by a horizontal translation two units to the right. Now, of course, our function is sin 𝑥. So we need to identify the function 𝑓 of 𝑥 minus two. This will give us the graph of the function that we were given. And of course, to achieve this, we replace the value of 𝑥 with the expression 𝑥 minus two. So 𝑓 of 𝑥 minus two is sin of 𝑥 minus two.
We now know that sin of 𝑥 maps onto sin of 𝑥 minus two by a horizontal translation two units to the right as required. And so the correct answer to this question is (E). The graph is given by 𝑦 equals sin of 𝑥 minus two.
