Graph each function over a one-period interval.

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Question

Graph each function over a one-period interval.

y = csc (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 function is Y equals the cosecant of X minus three pi divided by four. Then we are given a blank graph. We've 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 2 pi and then it's labeled in increments of pi divided by four. All right, we recall from previous lessons that when we are graphing cos can't, we are going to start by graphing its reciprocal function and its reciprocal function is the sine function. So we'll start by graphing Y equals the sine of the same thing X minus three pi divided by four. The first thing we want to do in graphing is we take a look at our B value, that's what's being multiplied directly to our X. And here there's nothing. So there's an invisible one. And when the B value is one, that means that our period is going to be two pie. So we're going to make an XY table and for our X values, we'll start with zero. And then we want to divide two pi into four equal parts. We recall from previous lessons that two pi divided into four equal parts is pi divided by two pie. Three pi divided by two and two pi. Then we're going to plug those into our sine function to find the Y values Y equals the sine of our X values minus three pi divided by four. So at this point, you can plug those into a calculator. If you plug in zero for the X value, you're going to get a Y value of negative square root two divided by two which is approximately negative 0.7. For the X value of pi divided by two, you'll get a Y value of negative square root two divided by two. Again, approximately negative 0.7. For an X value of pi, you'll get a Y value of positive square root two divided by two, approximately positive 0.7. For the X value of three pi divided by two, you'll get a Y value of square root two divided by two. Again, approximately positive 0.7. And for the X value of two pi you'll get a Y value of negative square root two divided by two, approximately negative 0.7. So if we plot these points on the graph, the 1st 10, negative 0.7. That's approximately here. The next point is pi divided by two negative 0.7. The next point is pi 0.7. The next point is three pi divided by 20.7. And then the last point is two pi negative 0.7. And so it kind of looks like a sign function. But here we can't see where our X intercepts are, we can't see our maximums and minimums. So what do we do? Well, what we can do is take a look at what's going on with our original function here is the X minus three pi divided by four. That is a phase shift to the right three pi divided by four. And because that's not in our increments of pi divided by two pi three pi divided by two and two pi it's going to be off a little bit as we can see. So what we can do is create a few more XY values to plot and we'll base our X values on three pi divided by four. So we can use X values of pi divided by four, three pi divided by four, five pi divided by four and seven pi divided by four. And when we plug those in to our Y equals the sine of X minus three pi divided by four. For pi divided by four, our Y value is going to be negative one for three pi divided by four, our Y value is zero for five pi divided by four, our Y value is one and for seven pi divided by four, our Y value is zero. Now we can plot those points pi divided by four negative one, three pi divided by 40 five pi divided by 41 and seven pi divided by 40. Now there we can see a much clearer sine function trying to connect those dots as smoothly as possible or more smoothly than me. Alright. Thats not too bad. And then now we can work off of this sine function to build our cos can graph, we recall from previous lessons whenever we have an X intercept for our sine function, our KC can is going to have a vertical Asymptote. So we have vertical asymptotes at X equals three pi divided by four and another vertical Asymptote at X equals seven pi divided by four. Next, wherever our sine function is below the X axis or KANT is also below the X axis. And it's going to come up in intersect where our sine function has a minimum. So along from the interval from X equals zero to X equals three pi divided by four at X equals zero. If you were to plug that into your calculator, if you had one divided by the sign of and then you had zero for the X value minus three pi divided by four, you'll get approximately negative 1.41. So that's our Y intercept is going to be at zero, negative 1.41. And then from there, it's going to come up, it's going to intercept at that point of pi divided by four, negative one. Then it heads back down toward negative infinity for the Y values along the X intercept of three pi divided by four. Now from X equals three pi divided by four to X equals seven pi divided by four. Our sine function is above the X axis. So is our cos can function. It's heading toward positive infinity for the Y values along the vertical Asymptote, X equals three pi divided by four. It's going to come down, it's going to hit that point at five pi divided by 41, then turn around and head back toward positive infinity for the Y values along the vertical Asymptote of X equals seven pi divided by four. Now to the right of seven pi divided by four, it's heading down toward negative infinity along that vertical Asymptote. And then it's going to come back up and it's not quite going to reach that point to get all the way up to our sine function. And if you were to plug into your calculator, an X value there of two pi into the one divided by the sine of two pi minus three pi divided by four, it's again going to be a negative 1.41 where we reach the end of our period. So there is the function Y equals the kant of X minus three pi divided by four. Well done. We'll catch you on the next one.

Graph each function over a one-period interval.

y = csc (x - π/4)

Key Concepts

Cosecant Function

Phase Shift

Graphing Trigonometric Functions

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