Each function graphed is of the form y = c + cos x, y = c + si...

Question

Each function graphed is of the form y = c + cos x, y = c + sin x, y = cos(x - d), or y = sin(x - d), where d is the least possible positive value. Determine an equation of the graph.

Accepted Answer

Welcome back everyone. In this problem. If the trigonometric function shown as the possible forms of Y equals D plus sine, XY equals D plus Xy equals the sine of X minus C or Y equals the cosine of X minus C. What is the equation that represents the graph note that C should have the smallest possible positive value for our answer choices. A says it's Y equals a half cos Xby equals a half sine xcy equals the sine of X minus a half of a quarter of pi and Dy equals the core sine of X minus a quarter of pi. Now, in order for us to figure out which one of those functions it is. First, let's take a look at our graph what's going on here? Well, by looking first, we can note that it resembles the graph of Y equals the cosine of X. If I were to plot it on our graph here, recall that Y equals the cosine of X would look something like this. Let me use a different color here. So it would look something like this. OK? Something like that. So notice that both of them kinda look the same. But when we compare them, you can tell that this graph, OK. This graph on our, on our diagram here is shifted 1/4 of pi units to the right instead of starting at zero on the X axis, it starts, its peak is at 1/4 of pi. So from our graph, we can tell that it's the graph of Y equals the cosine of X translated to the right by 1/4 of pi units. It's a horizontal translation. Now, how do we represent a horizontal translation in a trigonometric function? We recall, OK. Recall and recall that for any trigonometric function Y equals F of X, then Y equals F of X minus F of X minus C represents a horizontal translation, horizontal translation by sea units. And if C is positive, it's moving to the right. If C is negative, that means it's moving to the left. So here, since our horizontal shift is 1/4 of pi units, that means C equals 1/4 of pi. And if C equals 1/4 of pi, then our function is actually equal to the cosine of X minus 1/4 of pi. If we substitute it into our general form, this equation would represent a shift of the cosine of X by 1/4 of pi units to the right, which is what is going on here. When we look on our answer choices that would have been answer choice T so this must be the equation or the function that represents the graph. Thanks a lot for watching everyone. I hope this video helped.

Each function graphed is of the form y = c + cos x, y = c + sin x, y = cos(x - d), or y = sin(x - d), where d is the least possible positive value. Determine an equation of the graph.

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Key Concepts

Amplitude and Vertical Shift in Trigonometric Functions

Phase Shift in Sine and Cosine Functions

Identifying the Function Type from Graph Characteristics

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