Determine the amplitude, period, and phase shift of each funct...

Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.

y = 4 cos(2x − π)

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. Given the function Y equals 2 multiplied by cosine of 4 X minus 3 pi, identify the amplitude and phase shift from the options below. Then sketch its graph by considering only one period. Awesome. So it appears for this particular problem we're asked to solve for 4 separate answers. Firstly, we're trying to figure out the amplitude, then we need to figure out the period, and then we need to figure out the phase shift. And then our last answer we're trying to ultimately solve for is we're trying to figure out how to sketch this particular function as a graph considering only one period. OK. So with that in mind, let's read off our multiple choice answers to see what our final answer set might be, noting we're going to read the amplitude first, then the period, then the phase shift. So A is 2, pi divided by 2, and 3 pi divided by 4, B is 2, and pi divided by 2, and negative 3 pi divided by 4. C is 32 and 3 divided by 4. And finally, D is 32 pi and negative 3 pi divided by 4. Awesome. So first off, we need to note that our given function that's provided to us by the prom itself is written in the following form, which, as you should recall, Y equals A, multiplied by cosine of BX minus C. Fantastic, where capital A denotes the amplitude, and we need to take the absolute value of the amplitude. And the amplitude in this case, following this format, that means that 2 represents A, so we need to take the absolute value 2, which will just give us 2. Now we need to solve for the period P, and the period P is equal to 2 pi divided by B, which our value for B in this case is 4, so we need to take 2 pi divided by 4, which will give us. I divided by 2. And finally for the phase shift, we could solve for the phase shift by taking C divided by B. And when we take C divided by B, we will get 3 pi divided by 4. Awesome. So now our next step is we need to graph our functions. So when we go to grapher function, we need to start with the phase shift. And as we should note, X1, or it's X subscript 1, so X1 is equal to 3 pi divided by 4 because that's our initial phase shift, that's the value that we found a moment ago. Awesome. So now we need to solve for a few more points, so to keep it even to make sure we have a nice curve, let us solve for 5 different values of X so we can plot it on our graph to have a nice curve. So let's solve for X2, so we're going to be solving for X1, X2, X3, X4, and X5. But let's take a moment here to quickly recall and remember that we need to divide the period into 4. So let's make a little note here off to the side here, that the period divided by 4 is going to be equal to I divided by 2, divided by 4, which will mean that this is equal to I divided by 8. Awesome. Ultimately. So now we need to take this information and plug it into our expressions to solve for the different values for the phase shift. So to solve for X2, we need to take that X2 is equal to X1 plus our period divided by 4. So when we plug in our known variables, we know that X1 is equal to 3 pi divided by 4, and then we need to add pi divided by 4, which we determine that, that, sorry, I said pi divided by 4. We determined that divided by 4 is going to be equal to plus. Pi divided by 8. So, let me plug that into our calculator because once again, our period divided by 4 is going to be equal to pi divided by 8. So when we plug that into our calculator, we will get 7 pi divided by 8. And similarly, we could solve for X3, X4, and X5 in a similar fashion. So X3 is equal to X2 plus our period divided by 4, which the skip, I'm gonna skip over, you know, writing it out, but when you plug in your known variables, you will find that X3 is equal to pi. Then X4 is equal to X3 plus the period divided by 4, which is equal to 9 pi divided by 8. And then finally, 5 X is equal to 4 X plus your period divided by 4. And this is going to be equal to 5 pi divided by 4. Awesome. So now our next step is now that we have a, we have 5 different points we could plot, let's create a table and use our table to help us plot the different coordinate like coordinate points on our graph to help us sketch our graph more effectively. So with that said, Looking at our different values of X that we just found together, so we have for X we have 3 pi divided by 4, we have 7 pi divided by 8, we have pi and 9 pi divided by 8 and 5 pi divided by 4. And when you plug in these different values of X into our function to solve for Y, you will find that Y will equal 20, -2, 0, and 2 respectfully. And as we should recall and note, we should follow the following coordinate set up to help us plotter points, which is X. Y. So if X is positive, you go to the right, and if X is negative, you go to the left on the horizontal X axis. And if Y is positive, you go up following the vertical Y axis. You go up if it's positive or you go down if it's negative. So as you can see when we plot all five of our points. We will have a nice little curve, and since we're considering that it's only one period, so that means it since it's very skinny, so it's like it goes up and then it dips down and then it goes up again, so that's one curve. So we got it to 1 period, which is awesome. So that means at this particular point, we have officially solved for this problem. Wow, we did it. So now it's important to note that looking at our multiple choice answers, the correct answer has to be multiple choice answer, letter A, which once again is amplitude is equal to 2, is equal to pi divided by 2. And the face shift is equal to 3 pi divided by 4, and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.
y = 4 cos(2x − π)

Key Concepts

Amplitude of a Trigonometric Function

Period of a Trigonometric Function

Phase Shift of a Trigonometric Function

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