In Exercises 29–44, graph two periods of the given cosecant or se...

Question

In Exercises 29–44, graph two periods of the given cosecant or secant function.

y = 3 csc x

Accepted Answer

Welcome back. I am so glad you're here. We're asked to sketch the graph of the following Cosic can function using three periods. Our CO C can't function is Y equals four Cosic can X? All right. How do we graph Cosic can? Well, we want to begin by graphing its reciprocal function which is going to be the sine function. So we're going to start by graphing Y equals four sine X. And so we'll spend quite a bit of time doing that. Now looking at this Y equals four sine X, we see that's in the format of Y equals a sine of B X where A is what's being multiplied by the sign A here is four and B is what's being multiplied by the X. That's just an unwritten one. So now we're going to use that information for step one of graphing that we recall from previous lessons. And from the steps in the textbook in step one, we want to figure out the amplitude, the period if there's a phase shift where it begins. And because we don't have a graph made out for us, we want to figure out the domain and range so that we can draw our graph appropriately. So for step one, let's start with the amplitude, the amplitude is going to be the absolute value of a, well A is four. So it's the absolute value of four which is four. Now the period is going to be two pi divided by B. So two pi divided by R B here is 12 pi divided by one is just two pi. So our period is two pi now is there a phase shift? There is no C C is zero. So there is no phase shift, which means that for the sine function that we're graphing sign with no phase shift is going to begin at the origin. It's going to begin at 00. Now, for the domain we're going to begin at zero. But how far are we going? Well, we want three periods, a period is two pi. So three of those is going to be six pie. So we're going from 0 to 6 pie. How about the range? Well, the range for the sine function is going to be the amplitude above and below the X axis. So for the negative first, that's going to be at a negative four up to a positive for the amplitude below and then above the X axis. However, we're not just graphing the sine function. Our goal is to graph the CO C and the range for CO C is going to be from negative infinity to positive infinity. Now, as we're almost ready to start marking our graph up to draw that out, we want to know how to divide what units to put. And so for the range for the Y axis, because it's from negative 4 to 4, we can just mark those 01234 above and below. But for the domain, we're going to be using quarter periods for our key X values. So if we take a period and divide that by four, getting our quarter periods, a period here is two pi and dividing that by four simplifies two pi divided by two. So we can mark our X axis with units of pi divided by two. OK. Now we're ready to draw our graph because the domain is from 0 to 6 pi our vertical Y axis can be pretty far to the left and then the horizontal X axis can be right in the middle of that they meet at the origin. And now from the origin, we can label our Y axis, as we said, from the origin going up, we can go up and four is as high as our sign function is going to go, but the is going to go up to infinity. So we could label this, we can start with two, then four, then six and then eight that should give room for both functions. Now heading below the origin along the Y axis will have a negative two negative for negative six, negative eight. And now for the X values from the origin, we'll go to the right pi divided by two units all the way up to six pi. So from the origin, first marking is pi divided by two and adding pi divided by two to pi divided by two is going to be pi add pi divided by two is going to be three pi divided by two. Add pi divided by two is going to be two pi add pi divided by two is five pi divided by two. Add pi divided by two is three. Pi add pi divided by two is seven pi divided by two. Add pi divided by two is four. Pi add pi divided by two is nine pi divided by two. Add pi divided by two is five. Pi add pi divided by two is 11 pi divided by two. Now let me make a little more room. Add one more pie, divided by two is our six pie and that's as far as we're going and there we go, we've got this labeled and it's all set for us to start drawing our points. So now for this sign function step two step one had a lot of steps. Step two is going to be locating our five key X values that includes the X intercepts the maximum and the minimum. So X sub one, we already discerned that we're starting at the origin. So X sub one is zero, then X sub two, as we said, we're adding quarter periods to find each key X point. So this is already what we've identified along the X axis. So we're just adding pi divided by two to each previous X value. So for X of two, it's zero plus pi divided by two, which is pi divided by two, X sub three, we're adding pi divided by two. That's pi and we're just gonna keep adding pi divided by two so that we've gone three periods. So we'll have three pi divided by two. For X of four X of five is two. Pi X sub six is five pi divided by two X sub seven is three. Pi X sub eight is seven pi divided by two X sub nine is four. Pi X sub is nine pi divided by two, X sub 11 is five pie, X sub 12 is 11 pi divided by two and X sub 13 is our six pie. Step three is to discern the corresponding Y values. There are a couple ways to do this. We can plug our X values into our function Y equals four sine X that will get us our corresponding Y values. But we can also know just by understanding what goes on with our sine function, we said that this begins at the origin. So that would be 00. Our Y value here would be zero. Now every other of these is going to be an X intercept because A is positive our Y sub two, the X sub two and the Y sub two, that's going to be a maximum and the maximum is going to be the amplitude above the X axis. The amplitude is four. So it's going to be four units above the X axis. It's going to be four for our Y sub two. And as we said, every other one is an X intercept. So Y sub three is zero, Y sub four is going to be a minimum that's going to be the amplitude below the X axis. So four below is going to be a negative four. And then Y sub five, as we said, that's back to an X intercept back to zero. And now this cycle is just going to repeat itself. So Y sub six is back to four, Y sub seven is back to zero, Y sub eight down to negative four, Y sub nine back to zero, Y sub 10, back to positive four, Y sub 11 down to zero, Y sub 12 down to negative four and Y sub 13 back to zero. Now we can plot these points. So our first point is 00. That's at the origin. Second point is pi divided by 24 from the origin. We go to the right pi divided by two up four. Next point pi zero from the origin, we go to the right pi next point is three pi divided by two negative four from the origin. We go to the right three pi divided by two down four. Next 0.2 pie from the origin, we go to the right two pie. Next 0.5 pi divided by 24. From the origin. We go to the right five pi divided by two up four. Next 0.3 pi zero from the origin. We go to the right three pi next 0.7 pi divided by two negative four from the origin. We go to the right seven pi divided by two down four. Next 0.4 pi zero from the origin. We go to the right four pi next 0.9 pi divided by 24 from the origin. We go to the right nine pi divided by two up four. Next 0.5 pi zero from the origin. We go to the right five pi next 0.11 pi divided by two negative four from the origin. We go to the right 11 pi divided by two down four, last 0.6 pi zero from the origin. We go to the right six pi. Now we want to smoothly connect these this part. I'm so bad at this part. All right, we try to smoothly connect all of these. Oh my. Well, you can imagine that we're smoothly connecting these. Hopefully yours is more smooth. OK. So there is our sign function but we're not trying to graph the sign function. We're trying to graph coy can. So what do we do here is how we get to the CO C can function every X intercept for our sign function becomes a vertical asom tote Oops wanted to switch colors. OK. Becomes a vertical asom tote for the coy can every maximum of our sign function becomes a minimum or our coy can't function and every minimum or our sign function becomes a maximum where our CO C can't function. And with those three things, we can graph the whole thing. Let's begin with our X intercepts becoming vertical asom totes. We have an X intercept at 00 at the origin. So along X equals zero along that Y asom tote or excuse me, along the Y axis, we draw a vertical ASOM tote and label that X equals zero. Now the next X intercept is at pi zero. So we draw a vertical asom tote and label that X equals pi the next X intercept is at two pi so we draw a vertical ASOM tote through two pi zero and we label that X equals two pi the next X intercept is at three pi we draw vertical asom tote through three pi zero and label that X equals three pi the next X intercept is at four pi we draw vertical Asymptote through four pi zero and we label that X equals four pi the next X intercept is at five pi zero. So we draw the vertical ASOM tote through there, label that X equals five pi and our last X intercept is at six pi zero, draw vertical asom tote through there and label that X equals six pi. OK. Now our maximums become minimums and minimums become maximums. So for this to the right of our vertical Asymptote, X equals zero, we have a line that is up heading up toward positive infinity for the Y values along that asy tote, it's going to come down and it is going to curve and touch the sine functions maximum at pi divided by 24 that becomes the minimum for our co C can function and it turns around and heads back up to positive infinity for the Y values along the vertical asy tote X equals pi. There's a one line now for the next line, this is to the right of our vertical Asymptote X equals pi and this one is going to start in the negative infinity for the Y values. And it's going to come up along the vertical ASOM tote of X equals pi it's going to turn and touch the sine functions minimum at three pi divided by two negative four that becomes the new CO C can functions maximum turns and heads toward negative infinity along the vertical asom tote X equals two pi for the Y values. All right now to the right of our vertical Asymptote X equals two pi the Y values are heading up toward positive infinity. We draw it coming down along that vertical Asymptote, it's going to turn and touch the sine functions maximum at five pi divided by 24 that becomes the CO C can functions minimum heads back up to positive infinity for the Y values along the vertical asom tode X equals three pi. Now to the right of that vertical Asymptote, X equals three pi we're in the negatives for the Y values heading down toward negative infinity. We draw it coming up along that vertical asom tote it's going to turn and touch the sine functions minimum at seven pi divided by two, negative four becomes the co C can's maximum. Then it turns and heads back down toward negative infinity for the Y values along the vertical asom tote X equals four pi. Now to the right of the vertical Asymptote X equals four pi we're heading up toward positive infinity for the Y values. We draw it coming down along that Asymptote. It's going to turn and touch the sine functions maximum at nine pi divided by 24. And then it's going to turn and head toward positive infinity for the Y values along the vertical asy tote X equals five pi. Now to the right of the vertical Asymptote, X equals five pi we're down toward the negatives for our Y values. We draw it heading up along the vertical Asymptote, X equals five pi. It's going to touch the sine functions minimum at 11 pi divided by two negative four, turn around head toward negative infinity for the Y values along the vertical asom tode X equals six pi and that's it. That is our graph of Y equals four co can X well done. We'll catch you on the next one.

In Exercises 29–44, graph two periods of the given cosecant or secant function. y = 3 csc x

Key Concepts

Cosecant Function

Graphing Periodic Functions

Vertical Asymptotes

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