Graph each function over a two-period interval.
y = cos ...

Question

Graph each function over a two-period interval.

y = cos (x - π/2 )

Accepted Answer

Welcome back everyone. In this problem, we want to graph the function of Y equals the cosine of X minus 1/4 of pi over two periods. For our answer choices. You'll notice here that it shows all these graphs for two periods. And we're supposed to try and figure out which one is the correct graph of our function. Now, before we figure it out first, we need to think about our cosine function here. So let's compare our graph to Y equals the cosine of X. Now, what do you notice? Well here notice that the difference is in our function. We're subtracting 1/4 of pi from X. Now, what does that mean? Why is that important? Have you ever, have you ever seen anything like this before? Where something is changing inside the function? Well, recall that four Y equals F of X, eh If we have a trigonometric function written in a form F of X, then if we have a function that looks like that, why equals F of X minus C? It's going to represent a horizontal translation, horizontal translation of Y equals F of X or of our function by sea units. OK, we, if it is translated to the right C is going to be positive. And if C is negative, it's going to be translated to the left. So what we need to figure out, then we need to figure out which one of these or if we can figure out the value of C from our function, then we should be able to plot the cosine graph, shift it by that amount and then compare it to our answers. So first let's try to figure out the value of C and what it means. Now here, let me go down here and let me try and put it right here. We have some space, we have a graph that we can work with to compare. So by comparison, OK, if we compare Y equals a cosine of X minus 1/4 of pi two Y equals F of X minus C, we can notice here that both of them are in the same form. So then that the seed is going to be equal to 1/4 of pi. So C equals 1/4 of pi so that we know how much units we're going to shift it by no C is also positive. OK. C is positive. So because C is positive, that means we're going to shift it to the right. So this function represents a shift of the cosine function to 1/4 of pi units to the right. So if we were to draw the cosine function on our graph, then we should be able to shift it. So let's draw that graph. What does it look like? Well, we know that it's going to look something like this. So I'm gonna try my best here but we know it's going to look, it's gonna start at one. That's what our cosine graph looks like. Go to a half of pi and it goes to pi and then it comes back to three halves of pi goes to two pi, that's one period, right? Then we're gonna go to five halves of pie until we go down to three pie, come back to seven halves of pie. And then we go to full pie. I was so close. Oh jeez. No, I don't wanna do that. Let me take this off a bit. OK? And let me just fix that. So this is my sketch. OK. Fair enough, close enough. No, this is the graph of Y equals the cosine of X. And as we said, we're going to move it to the right by 1/4 of pi units. So that means instead of starting here at X equals zero, its peak is going to be here at X equals 1/4 of pi. So if I were to take the graph then and move it to that direction, I do remember that. I, I don't know that if I were to take the gun and move it here. No notice that its peak starts at 1/4 of pi and we can imagine because our graph is not piece wise. So it was continuing. So that means it would have been coming from the end here. So this is basically what our graph should look like. So let's basically look for a graph that starts the COSA graph at 1/4 of pi. Now, when we look at answer choices A and B notice that it doesn't have anything starting there. So we know that can't be right. And these represents functions either way. When we look at sea, notice that it has something nearby there, but it's not starting at 1/4 of pie, it's a bit higher. Therefore, logically, our correct choice must be answered choice D and we know that it is correct because when we look at it, it's starting at 1/4 of pi OK, starting at 1/4 of pi. So therefore our correct choice is answer choice. D. Thanks a lot for watching everyone. I hope this video helped.

Graph each function over a two-period interval.
y = cos (x - π/2 )

Key Concepts

Graphing Trigonometric Functions

Phase Shift in Trigonometric Functions

Period of the Cosine Function

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