Graph each function over a one-period interval.

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Question

Graph each function over a one-period interval.

y = - (1/2) csc (x + π/2)

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 negative 1/4 Kent of X plus three halves pie. And for our answer choices, we have four different graphs. We will look at the four graphs in detail in a little bit. But first, let's move this to a different part of the screen and we will work on building our own graph going to pull down our function. And here we've got a blank graph that will start to fill in. 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 three to positive three. And the domain for what's drawn for our X axis is from 0 to 5 pi divided by two. And it is marked in increments of pi divided by two. All right. So taking a look at our given Y equals negative 1/4 kant of X plus three halves. Pi we recall from previous lessons that K can't is the reciprocal function of the sine function. So to start our graph, we're actually going to start off graphing the sine function Y equals negative 1/4 sign of X plus three halves pie. And so here we can see that RB the value that's being multiplied by our X is one. And so our period is going to be two pie. We recall that from previous lessons. And so what we want to do is start off creating an XY table and we're going to choose five values for our X starting with zero. And we're dividing the period up into four equal parts. So after the X value of zero, it would be pi divided by two pi three pie, divided by two and two pie. And then we're going to plug those into the equation Y equals negative 1/4 sign of X plus three halves pie and plugging those in, you could use a calculator. If we plug in for a zero or an X value of zero, we're going to get a Y value of a positive 1/4 we plug in an X value of pi divided by two, we get a Y value of zero. We plug in an X value of pi, we get a Y value of negative 1/4. If we plug in an X value of three pi divided by two, we get a Y value of zero. And if we plug in an X value of two pi, we get a Y value of 14. Now that's for our sign, let's go ahead and graph that first point is 0 1/4 fro so from the origin will go up one quarter of a unit. Next point is pi divided by 20. The next point is pi negative 1/4. The next point is three pi divided by 20. And the final point is two pi 1/4 we can connect those as smoothly as possible. And there's our sine function. But since we're graphing Koi can, now we need to switch gears into graphing Kos can wherever our sine function has an X intercept, that's going to be a vertical Asymptote. So at the point pi divided by 20, we can draw vertical asom coach. And then at the X value of three pi divided by two, we also have a vertical asom toach. And then for each of these three uh sections from the X equals zero to our vertical Asymptote of X equals pi divided by two. There, our Koi can function is going to be above the X axis because our sine function is also above the X axis. It's going to share that point of 0 1/4. And then our Kant is going to curve up toward positive infinity for the Y values along the vertical Asymptote of X equals pi divided by two. Now from pi divided by 2 to 3 pi divided by two, our sine function is below the X axis. So our CC can is going to be as well. It's going to head toward negative infinity along the vertical Asymptote X equals pi divided by two comes up, hits that point at pi negative 1/4 then comes back down and curves down toward negative infinity along the vertical Asymptote three pi divided by two. And now to the right of the vertical Asymptote three pi divided by two, our sine function is above the X axis. So our KC can will be as well. It's going to head toward positive infinity for the Y values to the right of our vertical Asymptote three pi divided by two. And then it's going to curve down an intercept at that point of two pi 1/4. And there it stops because we're only drawing for one period. So let's take some notes on this. It's got asymptotes at X equals pi divided by two. Let's scroll here we go. X equals pi divided by two and X equals three pi divided by two. And then it's in the first quadrant during the intercept or excuse me, the from zero two pi divided by two and from three pi divided by 2 to 2 pi and it's in quadrant or from pie divided by two, 23 pi divided by two, right. And those notes should help us determine which of our four answer choices are the correct. So now moving this to a different part of the screen and grab these notes. All right. So let's look at asymptotes. First, we want asymptotes lung pi divided by two and three pi divided by two answer choice. A has asymptotes lung pi divided by two and three pi divided by two. And now we want to see if it's in quadrant one from zero to pi divided by two and from three pi divided by 2 to 2 pi it is in this graph that looks great. And then also we want to make sure that it has those same points at 0 1/4. Pi negative 1/4 and two pi 1/4 and a hits all of those. So A's got potential. Let's see. Let's make sure that we can rule out the rest of them. Answer choice B has vertical asymptotes at pi divided by two and three pi divided by two. But from zero to pi divided by two, it's in quadrant four and we need it to be in quadrant one. So we can eliminate answer choice B for answer choice C we have vertical asymptotes at X equals pi divided by two, X equals three pi divided by two. And from 0 to 22 pi divided by two. We're in quadrant four. So we can eliminate answer choice. C answer choice. D we've got vertical asymptotes at pi and two pi instead of pi divided by two and three, pi divided by two. So we can eliminate answer choice D, all right. Answer choice A is the one that matches well done. We catch you on the next one.

Graph each function over a one-period interval.

y = - (1/2) csc (x + π/2)

Key Concepts

Cosecant Function

Graphing Trigonometric Functions

Transformations of Functions

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