Match each function with its graph in choices A - D.

Question

y = sec (x - π/2)

Accepted Answer

Welcome back. I am so glad you're here. We're asked to graph the following function and to consider only one period, our function is that Y equals the C can't of X minus three halves pie. And then we are given four graphs for our solutions. We are going to look at those four graphs in detail. At the end. For now, let's move this to a different part of the screen where we'll have a little bit more space to work. OK. So Y equals the C can of X minus three halves. Pi is going to be the reciprocal function of our cosine function. So that would be Y equals the cosine of X minus three halves pi. And so when we are graphing ac can't function, we want to first begin by graphing our reciprocal function, the cosine function. And in order to do that, we can start off by just creating an XY table. So for our X value, since we're only using one period, we can use zero pi divided by two pi three pi divided by two and two pi and then for our Y values, we know that's equal to the cosine of X minus three halves pi and we can just plug in our X values to solve for Y a lot of ways to do this. You could plug it into your calculator at this point. So for an X value of zero, our Y value is also going to be zero for an X value of pi divided by two. Our Y value is negative one for an X value of pi, our Y value is going to be zero for an X value of three pi divided by two, our Y value is one and for an X value of two pi our Y value is going to be zero. So now we can graph these points going to draw a vertical Y axis, a horizontal X axis, they come together at the origin and I've offset the Y axis a little bit to the right or excuse me, the left on our X axis for the range for what I'm going to draw for the Y axis. I'm going to draw a negative one below the X axis and a positive one above the X axis. And for the domain for our X axis, I am going to label our X values from our XY table. So we have a zero at the origin. We have pi divided by two, we have excuse me pie. Then the next one is three pi divided by two and the last one is two pi. So now labeling these values, these points from our table. The first one is at 00. The next point is at pi divided by two negative one. The next point is at pi zero. The next point is at three pi divided by 21. And the last point for one period is at two pi zero. Now, we can connect all of these with smooth lines or you can imagine that they're smooth and imagine that they're all connected. And that is our Y equals the cosine of X minus three halves pi So now we want to figure out the inverse function the C can't and we recall a few things from previous lessons for the C can't function wherever our cosine has a X intercept, then our C can't is going to have a vertical Asymptote. So we have a vertical Asymptote along the Y axis. We have a vertical Asymptote at X equals pi we have a vertical Asymptote at X equals two pi and then our CC A is going to share a point with our cosine function wherever Y equals one or negative one when it is graphed without any stretching, vertical stretching. So here it's going to share the point pi divided by two negative one and it's going to share the 0.3 pi divided by 21. And then between our vertical asymptotes from X equals zero and X equals pi we're going to be in the fourth quadrant. Our C can is heading toward negative infinity along the Y axis. Then it comes up and hits that shared point at pi divided by 20 turns around and comes back down toward negative infinity for the Y values along the vertical Asymptote X equals pi. Now between our vertical asymptotes from pi to two pi we're going to be up in the first quadrant. Our C can function is heading toward positive infinity for the Y values along our vertical Asymptote X equals pi comes down, touches the 0.3 pi divided by 20 or excuse me, one and then it heads back up toward positive infinity for the Y values along the vertical Asymptote of X equals to pi and there is our Y equals the C can of X minus three halves pi graft. So we've got a graft and now we just need to find the matching answer choice. So I'm going to shrink this a little bit and move it to a different part of the screen. It's going to be a mess for a moment. And so first, let's compare it to our answer choices for A and B. All right. So for a, it looks like our function is between our asymptotes of X equals zero and X equals pi it's in the first quadrant and that is not what we have. We have our C can't function down in the fourth quadrant. So we can eliminate answer choice. A answer choice B has our graph. And here it has the C can function in the correct quadrants and its asymptotes all look the same. The first Asymptote is at X equals zero. The next one is at X equals pi and the final one is at X equals two pi. So that one's promising, let's check out C and D before we declare B the vector. All right. Four answer choice C again, it has the graph in the wrong quadrants at the wrong times. And it looks like the vertical asymptotes are also wrong. It has a vertical Asymptote at X equals zero. But the next one is at X equals pi divided by two and then a final one at X equals pi so we can eliminate answer choice C and answer choice D has those same vertical asymptotes as answer choice C, they're allowing X equals zero, X equals pi divided by two and X equals pi. So we can eliminate answer choice D and declare answer choice B our vector. So well done, we'll catch you on the next one.

Match each function with its graph in choices A - D.

y = sec (x - π/2)

Key Concepts

Secant Function

Phase Shift

Graphing Trigonometric Functions

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