In Exercises 35–42, determine the amplitude and period of each fu...

Question

In Exercises 35–42, determine the amplitude and period of each function. Then graph one period of the function.

y = cos 2x

Accepted Answer

Welcome back. I am so glad you're here. We're asked to find the amplitude and period of the given function and to sketch its graph for one period, our given function is Y equals the cosine of 13 X. All right. Well, how do we graph cosine? Well, first, we recognize that this is in the format of Y equals a cosine of BX where A is being multiplied by the cosine here, nothing's written. So that means that A is equal to one and B is being multiplied by the X here, B is equal to 13. So that's very helpful in finding the amplitude and period, which coincidentally is the first step for sketching its graph. So the amplitude for our cosine function is going to be the absolute value of A. Here we said A is one. So the absolute value of one, which is one for the period that's going to be two pi divided by B. So this is two pie divided by 13, B is 13 and there's our period. So we've got the amplitude is one and the period is two pi divided by 13. A few other things. First step one, we want to figure out where does this begin first? Is there a phase shift? Well, when it's written as Y equals a cosine of BX, there's no phase shift, there's no C. So for a cosine function, it will begin at zero A and when A is one, it's going to begin at 01. So that's very helpful. A couple of other things we're not given a graph to start with. So let's figure out our domain and range for our graph. The domain is all of the possible X values. We said we're starting at zero and then we'll be going for one period and one period is two pi divided by 13. So the domain domain is from 0 to 2 pi divided by 13. And the range, the range is going to be our amplitude above and below the X axis for cosine function. So for below the X axis, one amplitude below that is one below. So that's going to start at negative one and go to positive one, which is one amplitude above the X axis. So we've got our domain in range and then it also helps to know what our markings for the X axis are going to be. We're going to be marking by quarter periods. That's the period divided by four and so a quarter period, our period is two pi divided by 13 and dividing that by four. And that simplifies to be a period or excuse me, pi divided by 26. So we have markings along the X axis. Every pi divided by 26 units will help us. So we can draw our vertical Y axis and it can be pretty far to the left because our domain starts at zero and then we can draw our horizontal X axis right in the middle of that they meet at the origin. And for our range, we can have from negative one. So we'll draw a negative one below the X axis on our Y axis and then to positive one. So one above the X axis on our Y axis and now for our quarter periods. So we can mark to the right of the origin. We'll have pi divided by 26 as our first marking. And then adding another pi divided by 26 to that is pi divided by 13, adding another pie divided by 26 to pie divided by 13. We get the pi divided by and adding another pi divided by 26. To that we get two pi divided by 13 which is one period. OK. So we've got this graph set up. Now for the graphing of this, we want to figure out step two is our five key X values which will be for our X intercepts and our maximums and minimums not five sub six X sub five. There we go. OK. And we said that our first X value is going to be at zero, cosine begins at zero A and so X sub one is zero. Now, each subsequent X value will be adding a quarter period to the previous one. And we already figured this out zero plus a quarter period which is pi divided by 26 is pi divided by 26. Add another pie divided by 26 and we have pie divided by 13. Add another pie divided by 26 and we have three pie divided by 26 and add one more pie divided by 26. And we have two pie divided by 13, which is one period. Great. We're only doing one period. So that's enough for our X values. Now, step three is to find the corresponding Y values. And there are a couple of ways to do this. One way is that we can take our X values and plug them right in to our original cosine function. So for Y equals cosine 13 X, we'd plug in a zero for X, we'd have Y equals the cosine of 13 multiplied by zero. So zero. And that's where we would get a positive one. But we can also just figure out these Y values by understanding what's going on with our cosine function. We already said it begins at 01. Now the next point Y sub two that's going to be an X intercept. So the Y value there will be zero. And now for Y sub three, when Y sub one was a maximum Y sub three is going to be a minimum. So it's going to be the amplitude below the X axis. So it will be negative one for Y sub three, Y sub four comes right back up to another X intercept. So that's zero. And why sub five, that's one period later, that's going to be the amplitude above the X axis. That's going to be the positive one. Now we just need to plot these points. So the first point is 01 from the origin, we go up one unit. The next point is pi divided by 26 0 from the origin. We go to the right pi divided by 26. The next point is pi divided by 13 negative one from the origin. We go to the right pi divided by 13 and down one. The next point is three pi divided by 26 0 from the origin. We go to the right three pi divided by 26. And the last point is two pi divided by 13. 1 from the origin. We go to the right two pi divided by 13 and up one. And now we just need to smoothly connect these points and there is our Y equals the cosine of X. Well done. We'll catch you on the next one.

In Exercises 35–42, determine the amplitude and period of each function. Then graph one period of the function. y = cos 2x

Key Concepts

Amplitude

Period

Graphing Trigonometric Functions

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