Graph each function over a one-period interval.
y = csc(...

Question

Graph each function over a one-period interval.

y = csc((1/2)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 cosecant of 1/4 X minus pi divided by two. And then we're given a blanket 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 10 to positive 10. And the domain for what's drawn for our X axis is from 0 to 10 pi in increments of two pi. All right. So we recall from previous lessons that when we are going to graph A CO C can function, we're going to start off by graphing its reciprocal function, the sine function. So we'll graph Y equals the sine of the same thing 1/4 X minus pi divided by two. And so the first thing we do is we want to figure out our period, the period is given by looking at that B value that's what's being multiplied directly to the X. And here our B value is 1/4. Our period is given to us by two pi divided by B. So here it's two pi divided by 1/4 which is eight pi. So our period is eight pi and now we're going to create an XY table, we're going to choose some X values, find the corresponding Y values and then plot those for our sine function. So for the X values, we start with zero and then we want to divide our period into four equal parts. So if we're dividing eight pi by four, that first part is going to be two P and then we're going to add that to itself. Our next X values four pi adding another two pi to that is six P and then we have eight P. So there are our X values, we're going to plug those into the sine function, the sine of 1/4 multiplied by our X values minus pi divided by two. And that will give us our corresponding Y values. If you plug those into the calculator for the X value of zero, you're going to get a Y value of negative one for the X value of two pi, you'll get a Y value of zero for the X value of four pi you'll get a Y value of one for the X value of six pi youll get a Y value of zero. And for the X value of eight pi you'll get a Y value of negative one. Now let's plot those. So the first one, first point is zero negative one. The next point is two pi zero. The next point is four P one. The next point is six pi zero and the last point is eight pi negative one. So we can try to connect those as smoothly as possible. And there's our Y equals the sine of 1/4 X minus pi divided by two. Now, we need to figure out our KC cans graph. Well, we recall from previous lessons that when we are graphing A KC can anywhere the sine function had an X intercept that's going to be a vertical Asymptote. So we're going to have a vertical Asymptote at X equals two pi and we'll have a vertical Asymptote at X equals six pi. Now, wherever our sine function is below the X axis, our K can will be as well. And wherever our sine function is above the X axis, our kosi can will be as well. So looking from 0 to 2 pi for X values, the sine function is below the X axis. Our co C can function is going to share that minimum of negative one. And then it's going to head down toward negative infinity for the Y values along the vertical Asymptote, X equals to pi between our vertical asymptotes at X equals two pi and X equals six pi. Our sine function is above the X axis. So is our koi can, it's heading toward positive infinity along the vertical Asymptote, X equals two pi and it's going to come down and it's going to share that point at four pi one, then it's going to turn around and head back up toward positive infinity for the Y values along the vertical asom tote X equals six pi. Now to the right of the vertical Asymptote, X equals six pi our KC can function is heading toward negative infinity along that vertical Asymptote and then it's going to come up and then it's going to share that point at eight pi negative one. And there's our graph of the KOC camp of 1/4 X minus pi divided by two. Well done. We'll catch you on the next one.

Graph each function over a one-period interval.
y = csc((1/2)x - π/4)

Key Concepts

Cosecant Function

Period of Trigonometric Functions

Phase Shift

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