Graph each function over a one-period interval.
y = (1/2...

Question

Graph each function over a one-period interval.

y = (1/2) csc (2x + π/2)

Accepted Answer

Welcome back. I am so glad you're here. We're asked to graph the following function consider only one period. Our function is Y equals 1/4 co can of four X plus pi divided by two. Then we are given a blank graph on which we can draw our graph. We have a vertical Y axis, a horizontal X axis. They come together the origin, the range for what's drawn for our Y axis is from negative two to positive two. And then the domain for what's shown for our X axis is from zero to pi divided by two in increments of pi divided by eight. All right. So we recall from previous lessons that when we are graphing a Koc can function, we are going to start off by graphing its reciprocal function which is the sine function. So we're going to graph Y equals 1/4 S four X plus pi divided by two. And so the first thing we're going to do is figure out what the period is. We look at our B value. That's the one being multiplied directly to the X here. Our B value is four. And we recall from previous lessons that the period is found by taking two pi and dividing that by B. So in this case, we're taking two pi and dividing that by four and two pi divided by four reduces to pi divided by two. Our period here is pi divided by two. Now we're going to create an Xyt ball and we are going to plug in some X values, put those into the formula for Y equals 1/4 of the sine of four X plus pi divided by two and then plot those points. Our first X value is going to be zero. Now the rest of the X values, we're going to split our pi divided by two into four equal parts. So dividing pi divided by two by four gets us pi divided by eight. And now we're going to add that pi divided by eight to each of our subsequent numbers. So first adding it to itself pi divided by excuse me, pi divided by eight plus pi divided by eight is pi divided by four. Add another pie divided by eight, that's three pi divided by eight and add another pi divided by eight gets us up to our pi divided by two. Now we can plug each of those X values into our 1/4 sine of four X plus pi divided by two for an X value of zero. That's going to get us A Y value of 1/4 for an X value of pi divided by eight that will get us A Y value of zero for an X value of pi divided by four that will get us A Y value of negative 1/4 for an X value of three pi divided by eight that gets us A Y value of zero and for an X value of pi divided by two that gets us A Y value of 1/4. So now we can plot these points. The first one is 0 1/4. Next point is pi divided by 80. The next point is pi divided by four negative one of four. The next point is three pi divided by 80. And the last point is pi divided by 2, 1/4. We can try to connect those as smoothly as we can. And there's our graph for one period of Y equals 1/4 of the sign of four X plus pi divided by two. Now, we can work on our Kant graph. We recall from previous lessons that wherever our sine function has an X intercept, that's going to be a vertical Asymptote for our KC can. So we'll have our first vertical Asymptote at X equals pi divided by eight. And then we'll have another vertical Asymptote at X equals three pi divided by eight. Now, wherever our sine function is above the X axis, so is our Koi can function. So from the X values of zero to pi divided by eight, our Kant is going to be above the X axis as well. So it's going to share that point at 0 1/4. And then Kosi can, is going to head up toward positive infinity along the vertical Asymptote X equals pi divided by eight. Now from the X values of pi divided by 8 to 3 pi divided by eight, our sine function is below the X axis. So our CC can will be as well along the vertical Asymptote pi divided by eight are X or excuse me, our Kant is heading down toward negative infinity for the Y values. It is going to come up and it is going to share that point at P di by four, negative 14 and then it's going to turn around and head back down toward negative infinity for the Y values along our vertical asom tote X equals three pi divided by eight. Now to the right of our vertical Asymptote X equals three pi divided by eight. The sine function is above the X axis. So the KC can will be as well. It's heading toward positive infinity for the Y values along that vertical Asymptote three pi X equals three pi divided by eight. Then it's going to come down and it's going to curve and then share that point at pi divided by 2, 1/4. And there is our graph of Y equals 1/4 Kant of four X plus pi divided by two. Well done. We'll catch you on the next one.

Graph each function over a one-period interval.
y = (1/2) csc (2x + π/2)

Key Concepts

Cosecant Function

Period of Trigonometric Functions

Phase Shift

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