Graph each function over the interval [-2π, 2π]. Give the...

Question

Graph each function over the interval [-2π, 2π]. Give the amplitude. See Example 1.

y = 2 cos x

Accepted Answer

Hello. Today we are going to be graphing the equation on the interval from negative two pi to positive two pi we will be using the graph to find the amplitude. So we are given Y is equal to six cosine of X. Before we start graphing this function, let's go ahead and graph the traditional function Y is equal to cosine of X. The traditional function Y is equal to cosine of X starts at zero comma one and has a period of two pi which means that the function will end add two pi comma one. It has a minimum value located at pi comma one and zero is located at pi divided by two and three pi divided by two. And the shape of the cosine function isn't that of a reverse bell curve. So this is one period of the standard function cosine of X. But what is different between the standard function and the one given to us? Well, if we take a look at the given function, the difference here is that we have a coefficient of six instead of a coefficient of one. What this means is that the coefficient of six will have a change on the amplitude. If we compare the given trigonometric equation to the general equation Y is equal to a multiplied by cosine of B multiplied by X minus C plus D. We can identify the amplitude because the amplitude will be equal to the absolute value of A. In this case, the A term is the coefficient in front of the cosine function which in this case will be six. So our amplitude for the given equation is going to be the absolute value of six which will simplify to six. What this does is that it changes the range of our function. The range of our function is going to be from negative six to positive six. Or in other words, the maximum and minimum values will be located six units above and below the X axis. Let's go ahead and apply these changes to our cosine function. So the normal starting point will be zero comma one. But since our function has an amplitude of six, the starting point is going to be located at zero comma six, the zeros are going to remain unchanged, but the endpoint will be located six units above the X axis. So the end of the first period will be two pi comma six and the minimum value will be located six units below the X axis. So the new minimum value for the cosine function will be at pi comma negative six, the zeros will remain unchanged. So this is going to be the shape of the period of the function from 0 to 2 pie. Now, what we want to do is you wanna graph the second half of the function going from zero to negative two pi while the starting point is going to remain the same at zero comma six. The end point of this period is going to be located at negative two pi comma six and the minimum value will be located at negative pi comma negative six. The zeros will remain unchanged and the zeros will be located and negative pi divided by two and negative three pi divided by two. And if we were to connect the dots and keep the same shape, this is going to be the second half of the function. And in total, this will be the shape of the function on the interval from negative two pi to positive two pi. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Graph each function over the interval [-2π, 2π]. Give the amplitude. See Example 1.
y = 2 cos x

Key Concepts

Amplitude

Cosine Function

Graphing Trigonometric Functions

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