Determine the simplest form of an equation for each graph. Cho...

Question

Determine the simplest form of an equation for each graph. Choose b > 0, and include no phase shifts.

Accepted Answer

Welcome back. I am so glad you're here. And we are asked to write an equation that represents the graph shown. The equation should be in simplest form and phase shift is not included. We're given a graph, we have a vertical Y axis and we have a horizontal X axis. They come together at the origin. The range for what's drawn for our Y axis is from negative two to positive two. And the domain for what's drawn for our x axis is from 0 to 9 pi divided by eight. The Y axis is in increments of one and the X axis is in increments of pi divided by eight. We have a function line on that graph and the function line is passing through some key points. I'm going to write down an xy table with some of those. It begins going through the Y axis at the 0.0. And then that's about we're gonna go with three halves or 1.5. Then it passes through the point pi divided by eight zero and it's curving continuing downward going through the point pi divided by four negative three halves and then it begins curving upward, it passes through the 0.3 pi divided by eight zero and then curves upward back up to a maximum at pi divided by two positive three halves. Now that's one period. Our graph is shown continuing through another period. I'm not going to write those down, but so this one continues down. Passes through the 0.5 pi divided by 80 continues down with a minimum at three pi divided by four negative three halves, curves back upward through seven pi divided by 80 and then finishes its second period at pi three halves. All right. So that's what we're given. And what we want to do is come up with an equation for this first, we want to figure out what type of function this is. And when we have these smooth waves going up and down, we're dealing either with a cosine function or a sine function. But which one? Well, if we had a typical no changes standard, either cosine or sine function, we want to look at how it interacts with our Y axis, the cosine function is going to have a maximum passing through the Y axis. If it doesn't have phase shifts, if it hasn't reflected over the X axis, it's going to go through the 0.01 the sine function has not a maximum or a minimum. If it's been flipped, passing through the Y axis, it has its mid line passing through the Y axis. And if there's no vertical shift up or down, it's going through at 00. So do we have a maximum or a minimum passing through the Y axis or is it the midline? And we can see here that the midline is actually going right through the X axis? So at the Y axis, that is a maximum. So we're dealing with a cosine function. So our cosine function we recall from previous lessons is given with the equation Y equals C plus A multiplied by cosine of B multiplied by the quantity of X minus D. And now what's important is figuring out our A, our B, our C and our D plugging those in and then we'll have our equation. So what do each of those mean ABC and DD starting at the bottom has to do with phase shift? And we were told in our instructions, phase shift is not included. So D is equal to zero here. Great. Continuing to work backward, the C value is going to indicate a vertical shift up or down. The B value has to do with the period. And the A value has to do with amplitude. And it's also going to tell us if there's reflection over the X axis. So let's start with amplitude and reflection over the X axis. It can help to compare this to our standard cosine function in a standard cosine function A is equal to a positive one B is equal to a positive one C is equal to zero and D is also equal to zero. And we recall from previous lessons and from the textbook that those xy points that, that one would be passing through, the first point would be 01. And then it would be passing through the point pi divided by two zero and then pi negative one, then three pie divided by two zero. And finally, it would complete one period at two pi one. Now that goes beyond what has been drawn for our X axis. But we can see if we connect these as smoothly as possible that our cosine function has been much compressed. So its period is going to be different and it is stretched vertically. So its amplitude is going to be different. Is there reflection about the, about the X axis? Well, we start off with a maximum at the Y axis and then it heads downward with both our given function and the standard cosine function. So there's no reflection over the X axis. So are a value is positive. But for amplitude, we can calculate that we recall from previous lessons, we're going to take one half multiplied by the absolute value of our maximum Y value. That's going to be a positive three halves minus the minimum Y value that's going to be a negative three halves. So inside the absolute value bars three halves minus negative three halves, it's going to be a positive six halves, the absolute value of positive six halves is six halves, multiplying that times one half is going to get us right back to three halves. So our amplitude, our is a positive three halves and our, a value is a positive, the three halves here. So we've got our A now, our B has to do with the period. And we can see our normal cosine functions period would be much longer than this one. We fit two periods into where half a period would go. So we can calculate this, we know that our period is for cosine function, it's going to be two pi divided by B but here one period, it begins at zero and goes all the way to pi divided by two. So a period here is pi divided by two and we can calculate our B, we can cross multiply there. So we'll have four pi equals B pi well, they're dividing both sides by pi our B value equals four. Now C indicates any vertical shift. So if we had a vertical shift, our midline would no longer be the X axis. But our midline is the X axis, the maximum and the minimum have the same distance to our midline. They're both three halves away from the midline. So our C value here is zero. Now we have our ABC and D, we can just plug those in and we'll have our equation. So we have Y equals C zero. We don't have to put anything for that plus A A is three halves multiply by the cosine of our B is four, multiplied by the quantity of X minus DD is zero, X minus zero. We don't have to write minus zero. So we'll just put four X and there's our equation of our given function in this graph. Well done won't catch you on the next one.

Determine the simplest form of an equation for each graph. Choose b > 0, and include no phase shifts.
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Key Concepts

Trigonometric Functions

Amplitude and Period

Vertical Shift

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