Find a trigonometric function

Question

Find a trigonometric function $f$ represented by the graph in the figure.

Accepted Answer

Hello there, today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Describe the following graph using a cosine function. Awesome. So for this particular problem, simply, we're asked to figure out what this following graph is using a cosine function. So we're trying to describe this provided graph using a cosine function. So that is what we're ultimately trying to do. So we're trying to figure out how to describe this graph using a cosine function. So with that said, let's take a moment here to investigate our graph that is provided to us by the problem itself. And we have a curve represented in red, and then we have several different coordinate planes represented in black by black dots. So we have three separate peaks that have been labeled. So we have on the far left of our curve, we have a peak at -3 pi divided by 4.5. We have a second peak at, and this is on the positive side of our coordinate plane, i divided by 4.5, and then finally, our third peak is 5 pi divided by 4.5. And as you can see, looking at our curve, it only intersects the Y intercept at one point and on the Y or Y intercept, I should say, on the Y axis. So this is where it intersects with the Y axis, and it's gonna be intersecting only once on this Y axis at 0.2. And we have also one. like one peak, which this is a bottom peak, so we have the max peak and then we have our low peak. So this is the this is the part of the curve that curves downwards into the negative coordinate plane. And this peak is labeled at negative pi divided by 4.1. So negative pi divided by 4.1. Awesome. So, and it looks like we have a couple different periods here. Represented in our curve too, so that's good to keep in mind. So now that we have a visualization of what's going on for this particular problem, let's take a moment here to read off our multiple choice answers to see what our final answer might be, noting that for A, B, C, and D, it states that Y equals some equation. So A is 3 multiplied by the cosine of 2 X minus pi divided by 2, all plus 2. B is 3 multiplied by the cosine of 2 X minus pi divided by 2, all minus 2. C is 3 multiplied by the cosine of 2 X plus pi divided by 2 all plus 2, and finally D is 3 multiplied by the cosine of 2 X plus pi divided by 2 all minus 2. Awesome. So our first step in order to help us solve this particular problem, we need to recall and consider the following function. We need to consider and recall that Y is equal to A multiplied by cosine of B X plus C plus D. Awesome. So we need to recall and remember and use this following. Basically like a dummy equation. So this is like the layout for the format that we're gonna be using to help describe our graph as a cosine function. So, with that said, we now need to recall and note that the minimum value of this particular function, and let's make a note of this off to the side here. So our minimum, once again, our minimum value of the function is going to be -1. And this is our min. And once again, the maximum value of the function is going to be 5. And this is gonna be our max, and let's draw a little arrow so we don't forget. So we also need to recall and note that our amplitude, which will denote as A, our amplitude A is going to be equal to, as we should recall, plugging in our known variables. 5 minus -1. And we need to note that this will be divided by 2, which will give us an answer of simply 3. So now, Our next step is we need to solve for D. Once again, D is some constant value, and to solve for D, let us recall a note that D is equal to -1 + 5. Divided by 2, which is gonna be equal to 2. And now we need to solve for our period, which we'll call P, and our period P is going to be equal to 5 pi divided by 4 minus pi divided by 4, which will be equal to simply pi. Awesome. And our next step now is we need to solve for B, so we should recall that B is going to be equal to 2 pi divided by our period, which will mean that this is going to be equal to 2 pi divided by pi, which will give us 2. Let us also note that our horizontal shift, which I'll call HS and our horizon horizontal shift is going to be equal to pi divided by 4. And now we need to recall and use our equation to solve for C, and C is going to be equal to negative B multiplied by our horizontal shift. And let's use a multiplication symbol so we don't get confused here. So it's gonna be negative B multiplied by our horizontal shift. So that will mean that C is going to be equal to -2, multiplied by pi divided by 4, which will mean that it's going to be equal to negative pi divided by 2. Fantastic. We are on a roll here. So now when we plug in using our dummy equation, so that's the format, so that's how we know what variables to plug in to help us properly put together our cosine function. So when we plug in all of our known known values into our equation to solve for Y, which once again. is giving us our expression for the cosine function. We will find that our final answer for our cosine function will mean that Y is equal to 3 multiplied by the cosine of 2 X minus pi divided by 2 + 2, and that is our final answer. Hooray, we did it. And this corresponds with multiple choice answer letter A, which once again is Y equals 3 multiplied by cosine of 2X minus pi divided by 2, all plus 2. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

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