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 = 2 sec x

Accepted Answer

Welcome back. I am so glad you're here. We're asked to sketch the graph of the following C can't function using three periods. Our C can't function is Y equals four C can't X. Then we have a graph, the graph has a vertical Y axis, horizontal X axis, they come together at the origin. And then the range of our graph is from negative 10 to 10. And the domain of our graph is from negative pi divided by 2 to 13 pi divided by two, right. So how do we graph AC can function? Well, we are going to start off by graphing its reciprocal function. And the reciprocal of C can is cosine. So we're going to spend a lot of time graphing Y equals for cosine X. And we recognize that this is in the format of Y equals a multiplied by cosine of B X A is being multiplied by the cosine. So our A here is equal to four and the B is being multiplied by the X. The B here is one. Now we recall from previous lessons on graphing cosine. And from the steps in the book first, we want to take a look and find out what our amplitude, our period and our phase shift are. So for the amplitude, that is going to be the absolute value of A A is four. So it's the absolute value of four, which is four, we have an amplitude of four, our period that is going to be two pi divided by B our B is one. So two pi divided by one. Well, that's just two pi our period is two pi is there a phase shift? Well, there is no C term. So the C term is zero. So there is no phase shift, which means that this is going to begin for the X value, it's going to begin at zero and for the Y value it's going to begin at A because this is cosine. So it's going to begin at A. So this is at 04. All right. Now step two for graphing our cosine is we're going to look for our X values. We want to find our X intercepts are maximums and are minimums for the cosine. So we'll start with X sub one. As we said, we know we're starting with an X sub one at zero. Now to find all of the subsequent X values, we are going to add the period divided by four or a quarter period. The period is two pi so two pi divided by four. Well, that simplifies two pi divided by two. So for X sub two we're taking zero plus pi divided by two. That's pi divided by two. For X sub three pi divided by two plus pi divided by two is pi for X sub four pi plus pi divided by two is +33 pi divided by two. And for X sub 53 pi divided by two plus pi divided by two is two pi which makes sense. We have gone one period. But for this one, they're asking us to figure out three periods. So we'll go two more cycles. So X sub 62 pi plus pi divided by two is five pi divided by two for X sub seven, add another pi divided by two and we get to three pi for X sub eight, add another pi divided by two and we have seven pi divided by two for X sub nine, add another pi divided by two and we get four pi there's two periods. Now for X sub 10 add another pi divided by two and we have nine pi divided by two for X sub 11, add another pi divided by two and we have five pi almost there for X sub 12, add another pi divided by two and we have 11 pi divided by two. Then for X subir add another pi divided by two and we have six pi that is three periods. OK. Step three. As we are going to find the corresponding Y values to match with those X values there's a couple of ways to go about doing this. So you can always take the X value, plug it in for X in our function Y equals four cosine X. So this one would be Y equals four cosine zero and we would get a Y sub one value of four. But we can also figure all of these out just by understanding what's going on with our cosine function. So we're starting off with a maximum for our cosine function for our Y sub two, that's going down our Y, our X sub two and Y sub two are an X intercept. So the Y value is going to be zero now it's continuing down and it's going to be a minimum which will be the amplitude below the X axis. So our Y sub three is going to be the amplitude which is four below the X axis. So it'll be negative four for Y sub four, we're back up to an X intercept. So the Y value will be zero for Y sub five. We are back up to our starting value for Y, we're back up to the amplitude above the X axis which is four. And you can get all of these values plugging in the X values into that function. Now the cycle is just going to continue. So for Y sub six, we're repeating the pattern, we go back down to zero for Y sub seven, down to negative four. Then it turns around Y sub eight back up to zero. Y sub nine, back up to four, four hour. I don't know if I have enough room on the screen. Let's move this a little bit. We can draw a line. Ok. Probably still don't have room. Why? Sub 10? We are going to zero for our Y sub 11. This one goes down to negative four for the Y sub 12 that's back to zero. And for the Y sub 13, that one is back up to the starting value at four. OK. So we've got those 13 points. Now what we do is we graph our cosine function and then keep in mind that we're not done because we're trying to graph the C can't function. So let's start with the 0.4 from the origin. We go up four units. Next point is pi divided by 20. So from the origin, we go to the right pi divided by two units. Next point is pi negative four from the origin. We go to the right pi and we go down four units. Next is three pi divided by 20. From the origin. We go to the right three pi divided by two. Next point is two pi four from the origin. We go to the right two pi and up four, next point is five pi divided by 20 from the origin. We go to the right five pi divided by two. Next point is three pi negative four from the origin. We go to the right three pi and down four, next point is seven pi divided by 20. From the origin. We go to the right seven pi divided by two. Next 0.4 pi four from the origin. We go to the right four pi and up four next 0.9 pi divided by 20 from the origin. We go to the right nine pi divided by two. Next 0.5 pi negative four from the origin, we go to the right five pi and down four, next 0.11 pi divided by 20 from the origin. We go to the right 11 pi divided by two. Next 0.6 pi four from the origin. We go to the right six pi and up four. And that is three periods of our cosine function. Now we try to smoothly connect these points maybe more smoothly than this and there OK. There's our cosine function. Now back to what we started with, we want our C C function. So everywhere that we have an X intercept for our cosine function that is going to be a vertical as some tote for our C can't function everywhere that we have a maximum for our cosine function, that's going to be a minimum for our cca function. And everywhere we have a minimum, I might be spelling minimum. It doesn't quite look like minimum mini mum for our cosine function. We will have a maximum for our cca function. So let's get started. We'll start off drawing our vertical asom totes. So for our X intercepts, our first one is at pi divided by two. So we draw a vertical ASOM tote through pi divided by two label that X equals pi divided by two. Our next intercept is at three pi divided by two, draw vertical asom tote and label that X equals three pi divided by two. Next X intercept is it five pi divided by two. We draw vertical asom tote and label that X equals five pi divided by two. Next X intercept is seven pi divided by two. So we draw a vertical asom tote and label that at X equals seven pi divided by two. Next X intercept is at nine pi divided by two. So we draw our vertical asom tote and label that X equals nine pi divided by two. The last X intercept on here is at 11 pi divided by two, draw our vertical Asymptote and label that as X equals 11 pi divided by two. And so we've got all our vertical asom totes. Now let's draw our lines considering our maximums and minimums. So starting at our starting 0.4, we are going to begin there and then it's going to curve upward toward positive infinity for the Y values along our vertical ASOM tote at X equals pi divided by two. Now, the next line is to the right of our vertical Asymptote X equals pi divided by two. It's going to head down toward negative infinity for the Y values we can draw it coming up along our vertical Asymptote X equals pi divided by two. It's going to touch our cosines minimum at pi negative four that becomes this C cans maximum there. And then it turns around heads toward negative infinity for the Y values along the vertical asom tote X equals three pi divided by two. Now to the right of X equals three pi divided by two will be in the positive part for the Y values. And so that's heading toward positive infinity along that ascent tote it comes down, it's going to touch that maximum at two pi four for the cosine and then it's going to turn around and come along our vertical asom tode of X equals five pi divided by two, heading toward positive infinity for the Y values. All right, to the right of our vertical Asymptote X equals five pi divided by two. We're down in the negatives for our Y values. So that comes up along our vertical Asom tode it's going to hit that minimum for the cosine function become a maximum for our C can function at three pi negative four. And then it turns around heads toward negative infinity for the Y values along the vertical Asymptote of X equals seven pi divided by two, just a few more to the right of X equals seven pi divided by two. We're in the positive part for the Y values heading up toward positive infinity and then coming down along our vertical asom tote, we're going to touch our cosines maximum at four pi four becomes a new minimum for our C can turns around and heads toward positive infinity for the Y values along our vertical asom tote of nine pi divided by two. Now to the right of the vertical Asymptote, nine pi divided at X equals nine pi divided by two. We're in the negative part for the Y values. We are going to hug up along that vertical Asymptote and we're going to touch our cosines minimum at five pi negative four. Turn around head toward negative infinity for the Y values along the vertical Asymptote of X equals 11 pi divided by two. Last line to the right of X equals 11 pi divided by two. We're heading toward positive infinity for the Y values. Now we're going to draw it coming down along that as some tote and it's going to hit our cosines maximum at six P four and we're going to stop. That will be a minimum for our C camp function. And that's where we stop. And there is the graph of Y equals four C. Can't 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 = 2 sec x

Key Concepts

Secant Function

Amplitude and Vertical Stretch

Graphing Periodic Functions

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