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