Video Transcript
Which of the following is the graph of 𝑦 is equal to the sin of 𝑥 over four minus one? Options (A), (B), (C), (D), and (E).
In this question, we are asked to identify the correct graph of multiple transformations of the sine function. To do this, we can note that our transformed function is of the form 𝑓 of 𝑏 times 𝑥 minus 𝑑. We could do this by eliminating options. However, we will do this by sketching a graph of the transformed function. We can do this by recalling that the transformed graph can be found by first horizontally stretching the graph of sin 𝑥 by a factor of one over 𝑏 and then translating the graph down 𝑑 units.
In our case, the value of 𝑏 is one-quarter. So, we stretch the graph of sin 𝑥 by a factor of one over one-quarter, which is equal to four horizontally. And the value of 𝑑 is one, so we then translate it vertically down one unit. Let’s start by sketching the graph of 𝑦 equals the sin of one-quarter 𝑥, that is, a graph of 𝑦 equals sin 𝑥 stretched horizontally by a factor of four.
We know that the 𝑥-intercepts will be stretched by a factor of four. So, the first positive 𝑥-intercept is at four times 180, which is equal to 720 degrees. This means that on our axes, we will only have the 𝑥-intercept at the origin. We can note that at 360 and negative 360 we will have the maximum and minimum outputs of the function at one and negative one, respectively.
We then need to translate this graph down one unit. We can do this by translating each of the key points down one unit as shown. This then gives us the following sketch, which we can see matches the graph given in option (D).
