Fill in the blank(s) to correctly complete each sentence.

Question

The graph of y = -2 + 3 cos (x - π/6) is obtained by shifting the graph of y = cos x horizontally ________ unit(s) to the __________, (right/left) stretching it vertically by a factor of ________, and then shifting it vertically ________ unit(s) __________. (up/down)

Accepted Answer

Welcome back, everyone. In this problem, we want to figure out which of the following statements correctly describes the procedure for obtaining the graph of Y equals negative 14 plus five, multiplied by the cosine of X minus 1/4 of pi from the graph of Y equals the cosine of X. For our answer choices. A says that we will translate the graph of Y equals the cosine of X by 1/4 of pi units to the right stretch it vertically by a factor of five and translate it by 14 units downward. B says we will translate the graph a quarter of pi units to the right compress it vertically by a factor of five and translate it by 14 units downward. C says we'll translate the graph a quarter of pi units to the right stretched vertically by a factor of five and translate it by 40 units upward. And D says we're going to reflect it across the X axis stretched vertically by a factor of five and translate it by 14 units upward. Now, when we look at all of our answers, it's basically asking us to figure out the following three things from our graph, all our graph moves left or right. OK. So our left, all right. And in that case, rather, we can write that as our phase shift because that tells us how our graph moves left. All right. Next, it wants us to find the amplitude from our equation. That is how much the graph is stretched or compressed vertically. And then it wants us to find the vertical shift. That is how much the graph moves up or down. So we need to try and figure out those things from our equation. Now, what do we know about the cosine equation? Well, recall that the cosine equation or every cosine equation is written in the general form of Y equals a multiplied by the cosine of BX minus C plus D. OK. Where A is our amplitude. OK. And B it helps us to find our period, the period of our graph in this case where the period for a cosine graph equals two pi divided by the magnitude of B OK. C helps us to find our phase shift because the phase shift of our graph is going to be equal to C divided by B and D. OK. D said that properly here. D tells us our vertical shift. That is how much our graph moves up or down. So essentially in this problem, what we're trying to do is to take our graph of the cosine of X and figure out how our equation relates to tell us how we're going to move it where, where we're going to put it for our equation. So if we can figure out the amplitude, the vertical shift and the phase shift as asked for in our, in our set of answers, then we should be able to tell how the graph is going to move. So let's compare our equation to the general form of the equation. No or create a time equation that we have tells us that Y equals negative 14 plus five multiplied by the cosine of X minus 1/4 of pi. OK. Now, if you look at both equations, you can notice that at, at first it doesn't seem like they're really in the same form. So let's try to rewrite our equation in the general form. So we can do that by uh you know, putting our cosine function first. So that's five multiplied by the cosine of X minus of fourth of pi. And then we'll put our constant which is negative 14. No, they're both in the same form. So we can compare now by comparison, right, by comparison, notice that the coefficient of our cosine is our amplitude or rather is our value for A. So here A equals five, OK. In this case, B is the coefficient of our X term and here we have X. So that means B must be equal to one C is our constant here. So in this case, C would be equal to 1/4 of pi right? And D is our constant outside of the cosine term which equals negative 14. Now that we have the values of ABC and D, we can find our phase shift, vertical shift and amplitude since A is five, that means our amplitude equals five. OK. Next, our phase shift is going to be equal to C divided by B. So C is 1/4 of pi B is one and 1/4 of pi divided by one is simply 1/4 of pi. Since it's positive, that tells us that our phase shift is 1/4 of pi units to the right. OK. Finally, our vertical shift equals DK and here D equals negative 14. So since it's negative, that tells us then that our vertical shift which is negative 14 is 14 units down words because remember we're looking at how it moves up or down. Therefore, for our graph, it's, it's moved 1/4 of five units to the right. OK. It's stretched vertically by a factor of five and it's translated four units downward. If we look back on our answer choices, that would have been answer choice. A thanks a lot for watching everyone. I hope this video helped.

Fill in the blank(s) to correctly complete each sentence.
The graph of y = -2 + 3 cos (x - π/6) is obtained by shifting the graph of y = cos x horizontally unit(s) to the , (right/left) stretching it vertically by a factor of , and then shifting it vertically unit(s) ____. (up/down)

Key Concepts

Horizontal Phase Shift in Trigonometric Functions

Vertical Stretching and Amplitude

Vertical Shift of Trigonometric Graphs

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