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 has the possible forms of Y equals D plus sine, XY equals D plus cosine of 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 the cosine of X minus pi by equals the sine of X minus pi D. Sorry cy equals negative one plus the cosine of X and DY equals one plus sine X. Now, before we figure out which equation it is, let's try to get an idea of what's going on here. Uh Let's try to understand our graph. Does it look like any graph? You know? Well, if you look at it, doesn't it look like a cosine graph? Think about it. Let's plot our cosine graph. OK. So our, our cosine graph, a cosine graph would look something like this. Ready? I know if I were to move my cosine graph here, don't you realize it basically looks like our cosine graph? It is our cosine graph But what's the difference note that it has been moved, it has been moved by pi units instead of starting at zero on the X axis or instead of its peak being at zero on the X axis, its peak is at pi on the X axis. OK. Do you, can you see that there? So no, we notice that our graph is the cosine graph shifted by pi units to the right. So it's a horizontal translation. What do we know about horizontal translations? We recall, OK that for F FX, for Y equals F FX, that is a trigonometric function, then Y equals F of X minus C F of X minus C is a horizontal translation. The horizontal translation OK? Of our graph by C units where C is positive, that means it's moving to the right. And if C is negative, it's moving to the left here since our graph is moving pi units to the right, that means C would be equal to positive pi. So since C equals pi if we substitute it into a general form for a horizontal translation, this tells us then that the equation for a graph is going to be equal to Y equals the cosine because it's the cosine function of X minus pi that will be the equation for our graph. This equation tells us that the cosine graph is shifted by units to the right. When we look back on our answer choices that would have been answer choice. A so answer choice A is the correct equation 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 and Horizontal Translation

Identifying the Base Function (Sine vs Cosine)

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