Graph each function over a one-period interval.

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Question

Graph each function over a one-period interval.

y = 3 sec [(1/4)x]

Accepted Answer

Welcome back. I am so glad you're here. We are asked to graph the following function consider only one period. Our function is Y equals four. She cant of one half X. And then we are given four graphs for our answer choices. We are going to look at each of these in more detail once we've drawn our own graph. So let's head to a different part of the screen where we'll have a little bit more space to work on this. We'll bring our Y equals four. She can't of one half X. And here we've got a blank graph. It's got a vertical Y axis, a horizontal X axis. They come together at the origin. The range for what's drawn for our Y axis is from negative 10 to positive 10. And the domain for X axis for what's drawn is from 0 to 5 pi and each unit of pi is marked. So at pi two pi three pi and four pi, all right. So we recall from previous lessons that C can't is the reciprocal function of cosine. So we're going to start off by graphing Y equals four cosine of one half X and here, we can see that this is in the format of Y equals a cosign of BX. And when we have a value for our B, that's being multiplied by the X here, our B is equal to one half that's going to affect our period. We are asked to graph for one period and a period is given by two pi divided by B. So if B is one half, it's two pi divided by one half, which is a period of four pi. Now, in order to graph that cosine function, we're going to start off creating an XY table and our X values are going to start at zero. And then we want to divide our period into four equal parts. So dividing four pi into four equal parts. Well, that's going to be zero pi two P three pie and then four pie. And to find our Y values, we're again, we're working on right now. Y equals for cosine of one half X. And to find those Y values just plug the X values into that equation. So plugging zero in for X will get a Y value of positive four plugging pi in for X, we are going to get a Y value of zero. Plugging two pi in. For X, we get a Y value of negative four plugging three pi in. For X, we get a Y value of zero and plugging four pi in for X, we get a Y value of four. So now we can graph these points for our Y equals four cosine of one half X. Our first point is at 04 on the graph. The next point is at pi zero. The next point is at two pi negative four. Next point is at three pi zero and the last point is at four pi four. Now we connect these as smoothly as we can or you know, you can imagine that it's connected smoothly and well, and there we have our Y equals four cosine one half X. Now, in order to go to our reciprocal function of C can't a few things wherever we have an X intercept, that's going to be a vertical Asymptote. So we've got a vertical Asymptote at X equals pi and we've got a vertical Asymptote at X equals three pi. Now, wherever our cosine function is positive, our C can't function is going to be positive too. It's going to be here in the first quadrant. So we have that point where our cosine is at a maximum at 04, we're going to have our C can function start there and then it's going to curve and head up toward positive infinity for the Y values along the vertical Asymptote of X equals pi. Now between our vertical asymptotes of pi and three pi, our cosine function is negative for the Y value. So that means our C can function is also going to be negative for the Y values, it's going to head toward negative infinity along the vertical Asymptote X equals pi and then it's going to curve up and it's going to touch our cosine function at two pi negative four turn around and then head back down toward negative infinity along the vertical Asymptote of X equals three pi. And then our final one to the right of our vertical Asymptote X equals three pi we're heading toward positive infinity for our Y values along that Asymptote. And then it's going to turn and intercept at that 0.4 pi four. And then because we're only drawing one period, that's as much as we're drawing. So from zero two P, we are in the first quadrant, right quadrant one, that's not a one quadrant one. And from pie, 23 P were in the fourth quadrant. Then back from three pie to four pie, we are back to quadrant one. So we take those notes. Now let's go take a look at our answer choices and choose the correct graph. So moving this to a different part of the screen, we look, we'll start off looking at A and B and with a, let me move, move our notes so that we can see everything. All right with A, we've got vertical asymptotes at pi and three pi, that's correct. And from zero to pi we're in the first quadrant, but from pi to three pi we're in both the first and the fourth quadrants. So that one kind of peaked at A Y value of three, which is not what we have. So we can eliminate answer choice. A. Now looking at answer choice B, that one has vertical asymptotes at two pi and six pi, we have vertical asymptotes at pi and three pi we can eliminate answer choice B and now we'll look at C and D, all right, four answer choice C, we have vertical asymptotes at pi divided by two and three pi divided by two. That is not correct, can eliminate answer choice C. And for answer choice, D, we have vertical asymptotes with pi and three pi and then we've got those points at 04 at two pi negative four and at four pi four, those are all correct. We're in the correct quadrants with our function. Answer choice D is the winner. Well, then we'll catch you on the next one.

Graph each function over a one-period interval.

y = 3 sec [(1/4)x]

Key Concepts

Secant Function

Period of a Trigonometric Function

Vertical Stretch

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