Question Video: Identifying the Graph of the Sine Function | Nagwa Question Video: Identifying the Graph of the Sine Function | Nagwa

Question Video: Identifying the Graph of the Sine Function Mathematics • First Year of Secondary School

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Which of the following is the graph of ๐‘ฆ = sin ๐‘ฅ? [A] Graph A [B] Graph B [C] Graph C [D] Graph D [E] Graph E

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Video Transcript

Which of the following is the graph of ๐‘ฆ equals sin ๐‘ฅ?

Letโ€™s begin by recalling some of the important characteristics of the sine graph. Firstly, itโ€™s periodic with a period of 360 degrees, or two ๐œ‹ radians. So the same pattern repeats after every interval of 360 degrees. We know that weโ€™re going to be working in degrees in this question because looking at the five graphs, we can see that the values on the ๐‘ฅ-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 ๐‘ฅ-axis at every integer multiple of 180 degrees. In particular, ๐‘ฅ equals zero is the root of the sine function. And so the graph passes through the origin, or in other words the ๐‘ฆ-intercept of the graph is zero. We can now use these properties to identify which of the five graphs given represent ๐‘ฆ equals sin ๐‘ฅ. Graph (A) has a period somewhere between 90 and 135 degrees. So this is not the correct graph. It also has a ๐‘ฆ-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 ๐‘ฆ-intercept, but it has a period of 180 degrees. So this isnโ€™t 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 ๐‘ฅ-axis. The range of this graph is from zero to two, not negative one to one. And so we can rule out graph (C). Weโ€™re 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. Letโ€™s consider the roots of each function then. Graph (B) crosses the ๐‘ฅ-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 ๐‘ฅ-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 ๐‘ฆ-intercept of graph (E) is negative one and the graph doesnโ€™t pass through the origin. Graph (B) has the correct period, the correct range, the correct roots, and the correct ๐‘ฆ-intercept as well as having the correct shape. So graph (B) is the graph of ๐‘ฆ equals sin ๐‘ฅ.

If we were to sketch the graph of ๐‘ฆ equals sin ๐‘ฅ onto the same axes as graph (E), we would see that graph (E) is in fact a translation of ๐‘ฆ equals sin ๐‘ฅ. Each point has moved 90 degrees to the right. So we could say that graph (E) is a horizontal translation of ๐‘ฆ equals sin ๐‘ฅ by 90 degrees in the positive ๐‘ฅ-direction. The correct graph though is graph (B).

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