Video Transcript
Which of the following is the graph of ๐ฆ equals sin ๐ฅ?
Letโs begin by recalling some of the important characteristics of the sine graph. Firstly, itโs periodic with a period of 360 degrees, or two ๐ radians. So the same pattern repeats after every interval of 360 degrees. We know that weโre going to be working in degrees in this question because looking at the five graphs, we can see that the values on the ๐ฅ-axis are the integer multiples of 90. Secondly, the range of the sine function is the closed interval from negative one to one. The function oscillates continuously between its minimum value of negative one and its maximum value of positive one.
Next, the roots of the sine function are all the integer multiples of 180 degrees. So the graph of the sine function crosses the ๐ฅ-axis at every integer multiple of 180 degrees. In particular, ๐ฅ equals zero is the root of the sine function. And so the graph passes through the origin, or in other words the ๐ฆ-intercept of the graph is zero. We can now use these properties to identify which of the five graphs given represent ๐ฆ equals sin ๐ฅ. Graph (A) has a period somewhere between 90 and 135 degrees. So this is not the correct graph. It also has a ๐ฆ-intercept of one rather than zero. So this is a further reason why graph (A) does not represent the sine function.
Graph (D) does have the correct ๐ฆ-intercept, but it has a period of 180 degrees. So this isnโt the correct graph either. Graph (D) could represent a horizontal stretch of the sine function with a scale factor of one-half. Looking at graph (C), we can see that this graph sits entirely above and on the ๐ฅ-axis. The range of this graph is from zero to two, not negative one to one. And so we can rule out graph (C). Weโre left with graphs (B) and (E), which have the same shape. And we can see that they both have the correct range of negative one to one. Each graph also has a period of 360 degrees. Letโs consider the roots of each function then. Graph (B) crosses the ๐ฅ-axis at zero, 180 degrees, 360 degrees, and negative 180 degrees, negative 360 degrees. These are the integer multiples of 180 degrees. So graph (B) has the correct roots.
On the other hand, graph (E) intersects the ๐ฅ-axis at 90 degrees, 270 degrees, negative 90 degrees, negative 270 degrees, and so on. These are not the integer multiples of 180 degrees. And so graph (E) does not represent the sine function. We can also see that the ๐ฆ-intercept of graph (E) is negative one and the graph doesnโt pass through the origin. Graph (B) has the correct period, the correct range, the correct roots, and the correct ๐ฆ-intercept as well as having the correct shape. So graph (B) is the graph of ๐ฆ equals sin ๐ฅ.
If we were to sketch the graph of ๐ฆ equals sin ๐ฅ onto the same axes as graph (E), we would see that graph (E) is in fact a translation of ๐ฆ equals sin ๐ฅ. Each point has moved 90 degrees to the right. So we could say that graph (E) is a horizontal translation of ๐ฆ equals sin ๐ฅ by 90 degrees in the positive ๐ฅ-direction. The correct graph though is graph (B).
