Video: Figuring Out the Rule of a Trigonometric Function from Its Graph

What function is represented on the graph? [A] 𝑦 = cos π‘₯ [B] 𝑦 = 2 sin π‘₯ [C] 𝑦 = sin π‘₯

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Video Transcript

What function is represented on the graph? Is it A) 𝑦 equals cos π‘₯, B) 𝑦 equals two sin π‘₯, or C) 𝑦 equals sin π‘₯.

Even without looking at the options, just by looking at the graph itself, you could probably guess that this is a trigonometric function. The shape of the graph is sinusoidal. So it looks like either a transformation of sin or cos. Or of course, it could just be plain old sin or cos. Let’s call the function in the graph 𝑓 of π‘₯ and then find some properties of 𝑓 of π‘₯ which will help us find which of the functions in the options it is. If we look at the value of 𝑓 of π‘₯ when π‘₯ is zero, we can see that 𝑓 of zero is equal to one, just by looking at our graph. And immediately, we can eliminate options B and C. When π‘₯ is zero, both sin π‘₯ and two sin π‘₯ are also zero, and we want something which is one when π‘₯ is zero.

Our only remaining option, 𝑦 equals cos π‘₯, must therefore be the correct answer. Indeed, cos of zero is one. We can also see that the other features of the graph match up with what we would expect from the graph of 𝑦 equals cos π‘₯. The period of the function in the graph is two πœ‹ which is what we would expect from 𝑦 equals cos π‘₯ where π‘₯ is given in radians. The graph oscillates between a maximum of one and a minimum of negative one. Again, what we would expect. The 𝑦-axis is a line of symmetry of this graph. Another way of saying this is that the function the graph represents is an even function. And you may well know that cos of π‘₯ is an even function. So again, this tallies with what we know. All of this confirms the answer that we have.

The function represented on the graph is 𝑦 equals cos π‘₯.

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