Determine an equation for each graph.

Question

Accepted Answer

Welcome back. I am so glad you're here. We're asked to write an equation that represents the graph of the trigonometric function shown. And we have a graph that's got a vertical Y axis, a horizontal X axis. They come together at the origin in the middle. The range for what's drawn for our Y axis is from negative three to positive three. And the domain for what's marked for our X axis is from negative pi divided by six to positive pi divided by six. And that is in increments of pi divided by 12. We have two vertical asymptotes. One is at X equals negative pi divided by 12. And the other is at X equals a positive pi divided by 12. Then our graph of our function has three different parts to it. From the X values of negative pi divided by six to negative pi divided by 12. Our graph function is in the third quadrant. It's below the X axis. It has a point that negative pi divided by six and then, or excuse me, negative pi divided by six, negative one. And then from that point, it curves downward heading toward negative infinity for the Y values along the vertical Asymptote of X equals negative pi divided by 12 between the X values of negative pi divided by 12 to positive pi divided by 12. Our function is above the X axis in both the 1st and 2nd quadrants. It's heading toward positive infinity for the Y values along the vertical Asymptote, negative X equals negative pi divided by 12. It comes down in intercepts at the 0.01 and then it curves back up heading toward positive infinity for the Y values along the vertical Asymptote X equals pi divided by 12. The third section of our function that's drawn is between pi divided by 12 and pi divided by six for our X values. It's heading down toward negative infinity for the Y values along the vertical Asymptote pi divided by X equals pi divided by 12. And then it's heading up to the point pi divided by six negative one. And that's as much as is drawn, we have four answer choices. Our answer choices are answer choice. A Y equals the C can't of six X answer choice by equals the C can't of three X answer choice cy equals the KC can't of six X and answer choice DY equals the KC can't of three X. OK. So as we described, our function is existing where it has no X intercepts. It's got kind of U shapes heading toward either positive infinity for the Y values or negative infinity for the Y values. And it's got these vertical asymptotes. So that narrows down our functions with these U shapes we're talking about either AC can't or a cos can function. So now we recall from previous lessons and from the tables in the textbook that our c can't function. If it's not shifted, it's going to have a point at 01 and our CC can't function along that Y axis. It's going to have a vertical as some toach add X equals zero. So here, clearly, our function has a point at 01. It does not have a vertical Asymptote at X equals zero. So we're talking about ac can't function function. Now, our C can't function would take the format of Y equals a multiplied by the C can't of BX minus D. Let's talk about the A first A is going to give us vertical stretching or shrinking. If it didn't intercept at 01 and it was higher or lower along the Y axis, then that would indicate vertical stretching or shrinking. But because it's intersecting right at 01, then our A value is equal to one. So we don't have to put anything for it. It's a one. Now, let's take a look briefly at our devalue. Next, the devalue indicates a phase shift. So that would mean it's going to the right or to the left. Now, if it didn't intercept again at that spot, 01, that would indicate a phase shift. But that's right where the C can function is supposed to intercept the Y axis. So here our devalue is zero indicating there's no phase shift. Now, finally, let's talk about the B value, the B value is related to our period. Our period is given to us in the formula of two pi divided by the B value. And when the B value is one as would be typical with just a straight up C can function, then our period would be two pi two pi divided by one would be two pi. But here we can see a complete period because we're getting back to where we started with that value of A Y value of negative one at the points from negative pi divided by six to positive pi divided by six for our X values. So if we take our negative pi divided by six, and we'll take the absolute value of that plus our positive pi divided by six to find the distance between those two points, that'll show us our period. And that's going to equal pi divided by three. So that's not two pi our period here is pi divided by three. So we do have a B value. So what we do is we set our formula to pi divided by B equal to what our period actually is. Here pi divided by three, we can cross multiply to solve this multiplying two pi by three is six pi that's equal to pi divided by B or excuse me, pi multiplied by B which is B pi and here clearly six pi equals B pi B equals six. So we know that our B is equal to six and we can fill that in. We have Y equals our A value is one. We don't have to rate anything. The C can't of our B value is six. The X is given in there and then our D value is zero. So minus nothing, we don't have to put that here. The formula for the trigonometric function in the graph is Y equals the C can't of six X. We look at our answer choices and that matches with answer choice. A well done. We'll catch you on the next one.

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Determine an equation for each graph.