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 = -1 + cos x

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 represent the function Y equals negative eight plus the cosine of X. If we look down here, notice that we have all these graphs in answer choice ABC and D. Now if we're going to figure out which graph it is first, we have to think about what's going on here in our function. So let's compare it to the original cosine function, compare it to Y equals the cosine of X. What's the difference here? We notice that in our function it's subtracting it. The entire function is subtracting it. What does that mean? Well, recall that for a trigonometric function Y equals F FX. OK, then if we have a function Y equals F FX C. In other words, we're adding some constant adding or subtracting some constant to or trigonometric function, it's going to represent a vertical translation. So it represents a vertical translation of our original trigonometric function F of X. OK. Where if C is positive, it's an upward translation and if it's negative, it's a downward translation by C units. So if we can compare our general form for a vertical translation to our function, then we should be able to figure out how much it's moving by vertically. And thus which one of the graphs are correct. So let's go ahead and compare both. So I'm gonna make a note down here because here is where we, we can, we have an opportunity to sketch graph to get an idea of what it looks like. And by comparison, OK, by comparison, for Y equals negative eight plus the cosine of X compared to Y equals F FX plus or minus C. OK. OK. Here, there they are in the same form. It's just that one is written before the other. So I can actually turn my equation around, OK. Instead of minus eight plus the cosine of X, I can write it as the cosine of X minus it. You know that's the same thing and know what do you notice? Well, here we can tell that the value of CC equals negative it. So that means our graph is going to be vertically translated a units and C is negative. So since it's, it's, it's less than zero, it's going to be moving downwards. So Y equals the cosine of eight is cosine of X minus eight. Rather is the vertical translation of the graph of Y equals the cosine of X by eight units downward. So that means here if we were to try to sketch that on our graph just to get an idea of how that looks. We're talking about the cosine graph here and we know that our cosine graph looks something like this. OK? Cosine graphics, something like this. That's a complete period. And we notice we have our units here then to move it vertically downwards, to shift it downwards means we're going to move it from where it starts and it starts, its peak is at one here. Let me show that and we're going to move that eight units downwards. In other words, one minus eight, which would equal negative seven. So now it's going to start at negative seven. And when we move it, we move it all the way down. That means it's going to start right here. So that means our graph should be looking something like this. So let's go back to our answer choices and see if we can figure out which graph that is. Now when we look at our choices A and B are sine graphs, not cosine graphs. So we know they couldn't be correct. C and D are cosine graphs, but notice only C is showing a downward shift of eight units. In other words, our cosine graph or C starts at negative seven. So C has to be the correct answer. We know it can't be D because D is an upward shift, not a downward shift. Thanks a lot for watching everyone. I hope this video helps

Match each function with its graph in choices A–I. (One choice will not be used.)
y = -1 + cos x

A. <IMAGE> B. <IMAGE> C. <IMAGE>

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

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

Key Concepts

Cosine Function

Vertical Shifts

Graph Interpretation

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