Match each function with its graph in choices A–I. (One choice...

Question

Match each function with its graph in choices A–I. (One choice will not be used.)

y = cos (x - π/4)

A. B. C.

D. E. F.

G. H. I.

Accepted Answer

Welcome back everyone. In this problem, we want to figure out which of the graphs shown in our choices below represents the function Y equals the cosine of X minus a third of pi. And if we look on our answer choices, we notice that all of them represent some type of variation of different trigonometric functions. So we need to figure out which of them it is. Now if we're going to figure that out, we first need to think about what our function here means why equals the cosine of X minus a third of a third of pi let's compare it to of cosine function Y equals the cosine of X. What's the difference? What do you notice? Well, in our function, the difference is that it's subtracting a third of pi from X. What does that mean? Well, recall, OK. Recall that for any trigonometric function Y equals F of X. OK. Then if we have a function that's written in the form Y equals F of X minus C, in other words, a trigonometric function, but something is being subtracted from XR or angle, then that function is a horizontal translation of Y equals F of X or a horizontal translation of our trigonometric function. OK. And in that horizontal translation, then it's being translated by C units where if C is positive, then it's moved to the right. And if C is negative, it's moving to the left. So if we can compare our function to its general form, then we can figure out the value of C and its nature and thus eventually tell how much we are horizontally translating the cosine of X pi. So let's do that. Let's compare both of those. Now by comparison, let me put it right here because here's where we're going to draw our graph by comparison. Or function Y equals the cosine of X minus a third of pi compared to Y equals the F of X minus C. OK. Here we note that both of them are in the same form. So C must be a third of pi. So C equals a third of pi. So we know it's going to be translated by a third of pi units in which direction. Well, a third of pi is positive. So it must be moving to the right. Therefore, we can say then that our function is the horizontal translation is the horizontal translation off Y equals the cosine of X buy a third of pay units to the right. OK. I just want to put that there for a reference while we're enjoying it. So what this means is that we're going to draw the graph of the cosine of X and then move it a third of pi units to the right. So first, let's draw the graph of the coarse and of X. Now, what does that look like? But here, right, it essentially would look something like this. It starts here and then it goes down to half of pi. Well, let me draw that a bit more. Here goes down to pi and it comes back up and then it goes to two pi. That's a complete period. So what we're going to do is to move this a third of pi units to the right. So what would that look like? That means instead of starting at one, starting at zero here on the X axis, it would start at a third of pi, it would start somewhere right there. So I think I can move this and show that it starts right here. So that's the graph that we're looking for. If we look back on our answer choices here, notice that D is the cosine graph. Let me show everything D is the cosine graph and it starts at a third of pi that tells us then that D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Match each function with its graph in choices A–I. (One choice will not be used.)
y = cos (x - π/4)

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D. <IMAGE> E. <IMAGE> F. <IMAGE>

G. <IMAGE> H. <IMAGE> I. <IMAGE>

Key Concepts

Phase Shift in Trigonometric Functions

Basic Shape and Properties of the Cosine Function

Graph Matching Techniques for Trigonometric Functions

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