In Exercises 53–60, use a vertical shift to graph one period of t...

Question

In Exercises 53–60, use a vertical shift to graph one period of the function.

y = cos x + 3

Accepted Answer

Welcome back, everyone. In this problem, we want to sketch the graph of the function Y equals the cosine of X minus six for one period of the graph. Now, what do we already know? Well recall that for a trigonometric function, the graph is usually in the form Y equals a multiplied by the cosine of BX minus C plus D. Now, before we make sense of any of these variables, that's why I'm already loving this graph because it's just the regular graph of the cosine of X no phase shift or nothing. So A would be equal to one, of course, B would also be equal to one. So that tells us that the period is our same old period of two pi because the period is two pi divided by B. So that would be two pi divided by one, which is just two pi and our vertical shift is just negative six. So in layman's terms, what we're really doing is just we're going to sketch the cosine graph and then we're going to move it six units down. That's basically what we're doing. So let's do that. So let me first start by sketching my cosine graph here. So oops, that's my sine graph. Don't tell anybody. So my cosine graph is gonna look like this. OK? Where my period is from 0 to 2 pie. And I want to capture the important points. So halfway through his pie, which would be the minimum, uh halfway between that is half of pi where it cuts the axis, the X axis. And again, halfway between pi and two pi just a sketch that would just be three pi divided by two. So that's my cosine graph. I've already done half of the work. All I need to do now is to shift it down by six units. So I'm gonna take it here. I'm just gonna move it with my little tool here and I'm shifting it down by six units. So it's gonna probably be somewhere down here. OK? And no, I just want to see if I can capture the important points back on the graph. So let me do a little vertical line, uh a horizontal line for reference here. And I think I just wanna rub through it just to show you that this is just a reference line. OK? And no, let me see if I can put back my points. So we know we started here. So we're shifting it down by six units. So this would be our peak. Usually it would have been at one, but now we've shifted it down by six units. That's one minus six, which would be equal to negative five. So this point would be zero, negative five next at half of pipe. Then we know that oops that should be right here at half of pi we know it's gonna come back to our center. So that's gonna be half of pi and six for that point down here, it's our minimum. So it would have been one unit less. So that would be pi and negative seven. We come back to our baseline. So here it would have been three halves of pi and negative six. And then we come back to our peak, which would have been two pi and negative five because if you notice the peaks would have been the same. So no, this would be our sketch of Y equals the cosine of X minus six. Thanks for watching. I hope this video helped.

In Exercises 53–60, use a vertical shift to graph one period of the function. y = cos x + 3

Key Concepts

Cosine Function

Vertical Shift

Graphing Transformations

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