Determine an equation for each graph.

Question

Accepted Answer

Welcome back. I am so glad you're here. We are asked to write an equation that represents the graph of the trigonometric function shown. Then we are given a graph, we have a vertical Y axis, a horizontal X axis. They come together at the origin. The range for what's drawn for our Y axis is from negative 12 to positive five. And the domain for what's drawn for our X axis is from 0 to 5 pi divided by two. And that's in increments of pi divided by two. Then we have a function on here. We've got two U shapes, one's heading up and then one's heading down. There are vertical asymptotes at X equals zero, X equals pi and X equals two pi for those two U formations. The first one is between the vertical asymptotes, X equals zero and X equals pi that one's facing upward. And it has its lowest point at X equals pi divided by two and Y equals negative five. Now, the second U shape is heading facing downwards is between the X, the vertical asymptotes, X equals pi and X equals two pi and it has the top of its U shape at the 0.3 pi divided by two negative seven. And we are given four answer choices. Our answer choices are answer choice A Y equals negative six plus the C can't of X answer choice by equals six. C can't X answer choice, CY equals negative six plus the K can't of X and answer choice, dy equals negative five plus the KC cant of X. All right. So we've got these two U shapes and vertical asymptotes. And that's indicative of having either the C can't or the Kant functions. If we had AC can't function, we would have this intercepting with the Y axis. If it wasn't shifted up or down left or right, it would intercept at 01. And if we had a KC can't function, its interaction with the Y axis would actually be a vertical Asymptote. We would have an Asymptote at X equals zero. And again, that's if it's not shifted left or right, if there's no phase shift. And here, what we can see is there is a vertical Asymptote at X equals zero. So we're dealing with a CO C can't function, all right. A cos can function can be given to us with the equation Y equals a multiplied by the COS can of BX minus D. And then after that plus C. So we know the KC camp part, let's talk about the ABC and D parts. The A is going to indicate vertical stretching or shrinking and here there is no vertical stretching or shrinking this one, its use look just like it's supposed to. So our a value here is going to be one, the bee value is going to be related to the period. So normally, if we had just a straight up KC can't function, the period would be two pi and then the two pi gets divided by the B value. Here we have a complete period from the X values of 0 to 2 pi. So that is a straight up period of two pi, our B value is going to be one, the C value. I'm going to talk about in a moment. But first, I'm gonna talk about the devalue. The devalue indicates a phase shift left or right. Again, we wouldn't have had that vertical Asymptote right at X equals zero. If there had been the phase shift on the correct spots of X equals zero, X equals pi X equals two pi. So here there is no phase shift, our devalue is zero. But we do need to notice something about C here C indicates a shift vertically going up or down and this one has shifted vertically. So typically on our coast, he can't function. We would have nothing happening at the X axis. We wouldn't have any X intercepts, they wouldn't even come close. We would be up at the point pi divided by 21 and we'd be at the 0.3 pi divided by two negative one for the bottoms and tops of our es. But here we're down at pi divided by two, negative five and three pi divided by two negative seven. So we have shifted down six units. So our C value indicating those vertical shifts is expressed just by saying what's happening with the units because it's down, it's going to be negative and because it's six units, it's a six. So that's the only real change here. We can rewrite our form or our equation as Y equals A is one. We don't have to put anything cos can't of B is one, we don't have to write that X minus D is zero. We don't have to write that. And then after the parentheses plus CC is negative six. So we can write minus six. Now, another way to write that is to put the negative six in front of the KC can and that's the way they've done it for the answer choices. So the same thing is Y equals minus or negative six plus the KC can of X. And there is our equation for our given function. We look at our answer choices and this matches with answer choice. C well done. We'll catch you on the next one.

Determine an equation for each graph.

Key Concepts

Graphing Trigonometric Functions

Amplitude and Period

Phase Shift

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