Graph each function over a one-period interval.

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Question

Graph each function over a one-period interval.

y = sec (x + π/4)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to graph the following function consider only one period. Our given function is Y equals the C of X plus three pi divided by two. For our answer choices, we have four different graphs. We'll look at each of those graphs in detail once we've drawn our own for now, let's move this to a different part of the screen. Going to grab our function and bring it down to this blank graph here where we'll do our work for our blank graph. We have 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 three to positive three. And the domain for what's drawn for our X axis is from zero. And then it goes to five pi divided by two and it's marked in increments of pi divided by two. OK. We recall from previous lessons that when we are graphing the C can't function, we are going to begin by graphing its reciprocal function, which is the cosine function. So we're going to start off looking at Y equals the cosine of the same thing X plus three pi divided by two. Now, the first step is that we look at our B that's what's being multiplied directly to our X here. There's nothing. So that means there's an invisible one. And when that's the case, we recall from previous lessons that our period is two pie. So we're going to create an XY table and get some XY values and then plot those points for our X values. We take a look at the period and we want to divide that into four equal parts. We recall from previous lessons that when our periods two pi our X values are going to be zero pi divided by two pie. Three pi divided by two and two pie. And now we're going to plug those X values into our cosine function to find our Y values. So Y equals the cosine of X plus three pi divided by two. And now plugging in the X value of zero into the cosine of X plus three pi divided by two. That's going to give us a Y value of zero. Plugging in the X value of pi divided by two will get a Y value of positive one. Plugging in the X value of pi will get a Y value of zero plugging in the X value of three pi divided by two will get a Y value of negative one and plugging in the X value of two pi will get a Y value of zero. Now we can plot those points for our cosine 1st 0.00 at the origin. Next pi divided by 21, next pie zero, next three pi divided by two negative one. And last two pi zero, we can connect those as smoothly as possible and theres ry equals the cosine of X plus three pi divided by two. Now we recall from previous lessons that when we are going to graph the C can't, that means that wherever we have an X intercept for our cosine function, we're going to have a vertical Asymptote. So here we'll have a vertical Asymptote at X equals zero along the Y axis, we'll have a vertical Asymptote at X equals pi and we'll have a vertical Asymptote at X equals two pi and I'm going to make note of these for when we go back to look at our answer choices. So we have vertical asy totes at zero. Well, X equals zero, X equals pi and X equals two pi. OK. Next we know that wherever our cosine function is above the X axis, our C can function is also going to be above the X axis. So between the vertical asymptotes, X equals zero and X equals pi our C can function is above the X axis. It's heading toward positive infinity for the Y values along the vertical Asymptote, X equals zero. And it's going to come down, it's going to share that point pi divided by 21 that's going to turn back around and head back up to positive infinity for along the vertical Asymptote X equals pi. Now between the vertical asymptotes, X equals pi and X equals two pi our cosine function is below the X axis. So our C can't function is as well. It's heading toward negative infinity along the vertical Asymptote X equals pi it's going to head up, it's going to share that 0.3 pi divided by two negative one. Then turn back around, head back down toward negative infinity for the Y values along the vertical Asymptote X equals two pi. So we'll write down that from zero to pi we're in quadrant one and then we have that point at pi divided by 21 and then from three pie or excuse me, no, from pie to two P. We're in quadrant four and we have that point at three pi divided by two negative one. All right. So now let's go check out our graphs and we'll see which one which answer choice matches up with our drawn graph going to move this to a different part of the screen. Then I'm going to grab our notes and bring them up so we can see them with all of our answer choices. Let's start off looking at A A has vertical asymptotes at X equals zero X equals pi and X equals two pi. That's great from zero to pi it is in the first quadrant and it's got a point at pi divided by 21 great. And from X equals pi to X equals two pi. It's in the fourth quadrant fantastic. And then it has a point at three pi divided by two negative one. Answer choice. A looks great. Let's eliminate the rest of them. Answer choice. B has the correct vertical asymptotes at X equals pi X equals zero, X equals pi and X equals two pi. But from X equals pi to X. Excuse me, from X equals zero to X equals pi. It's in the fourth quadrant, not the first quadrant. So that is eliminated. Now lets look at answer choice. C answer choice C has vertical asymptotes at pi divided by four and three pi divided by four. That is not correct. Can eliminate answer choice. C answer choice D also has vertical asymptotes at pi divided by four and three pi divided by four. That is incorrect answer. Choice D is eliminated. Answer choice A did turn out to be the correct one. Well done. We catch you on the next one.

Graph each function over a one-period interval.

y = sec (x + π/4)

Key Concepts

Secant Function

Phase Shift

Graphing Periodic Functions

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