# Question Video: Identifying the Graph of a Phase Shift Applied to a Cosine Function Mathematics • 10th Grade

Which of the following is the graph of 𝑦 = cos(𝑥) + 1? [A] Graph A [B] Graph B [C] Graph C [D] Graph D [E] Graph E

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### Video Transcript

Which of the following is a graph of 𝑦 equals cos 𝑥 plus one?

So what we can see here is in fact 𝑦 equals cos 𝑥 plus one is gonna be a translation of 𝑦 equals cos 𝑥. And we know this is a translation because it’s in the form 𝑓 of 𝑥 plus 𝑎, where this is a shift in the 𝑦-axis of 𝑎. So we now we’re gonna have 𝑎 units shifted in the 𝑦-axis. But what’s it a translation of?

Well, what it is going to be is a translation of 𝑦 equals cos 𝑥. And we know what it’s gonna be is a shift in the 𝑦-axis of one unit. But what does this mean in practice? Well, what it means is we actually add one to each of our 𝑦-coordinates. Well, to help us work out which one of our graphs is going to be this shift of one unit in the 𝑦-axis, what we’ve done here is sketched 𝑦 equals cos 𝑥. And we’ve done it onto the graph (A).

Well, actually if we shift it one unit in the 𝑦-axis, so we add one to each of the 𝑦-coordinates, we can see that it’d map itself onto the graph that is shown in (A). Because instead of the peaks being at one, they would be at two. And instead of the troughs being at negative one, they would be at zero. So therefore, we can say that the graph (A) is the graph of 𝑦 equals cos 𝑥 plus one. Well, if we wanted to check the other graphs, we can see that these are incorrect because (B) is in fact the graph 𝑦 equals cos 𝑥. We’ve got (C), which is a phase shift; (D) is a period change; and (E) is a period change and a phase shift.