Graph each function over a two-period interval.
y = 1 - ...

Question

Graph each function over a two-period interval.

y = 1 - 2 cos ((1/2)x)

Accepted Answer

Hello. Today we are going to be graphing the following function and we will be graphing two periods of the function we are given Y is equal to two minus five cosine of X divided by three. Before we graph the function let's quickly rewrite the function to have the cosine function be the leading term doing so will allow us to rewrite the function as Y is equal to negative five cosine of one third multiplied by X plus two. So before we graph this function, there are four pieces of information that we need to identify. We need to identify the graph's amplitude, the period, the phase shift and the vertical shift. In order to get these pieces of information, we can com compare the giving function to the general function Y is equal to a multiplied by cosine of B multiplied by X minus C plus D. Using this general function, let's start by identifying the graph's amplitude. Now the amplitude is defined as the absolute value of A, this is where A is the coefficient in front of the cosine function. The coefficient in front of the cosine function is negative five. This means the amplitude is going to equal to the absolute value of negative five, which is equal to five. What this means is that the mixed ma maximum and minimum values are going to be five units above and below the X axis. Next, let's identify the graphs period. Now, the period for a cosine function is defined as two pi divided by the absolute value of B. This is where B is the coefficient in front of the X minus C term. That coefficient in this case is going to be one divided by three. This means the period is defined as two pi divided by the absolute value of one divided by three. And after algebraically simplifying this will give us six pi what this means is that the first period of the graph will start at zero and end at six pi. Next, let's go ahead and identify the face shift. Now the phase shift can be thought of as a horizontal shift of the graph. The phase shift is identified by the C term. Now, in this case, we are not given an X minus C term in the given function. What this means is that the C value is equal to zero. And that means we don't have a phase shift of the graph. Finally, we need to identify the vertical shift of the graph. Now the vertical shift is identified by the D term and the D term is any values that we are adding or subtracting to the cosine function. In this case, we are adding two units to the cosine function. What this means is that the vertical shift will equal to two. And that means we are going to shift the entirety of the function up to units now that we have these values. Let's go ahead and start graphing the function. So the traditional cosine function will start at zero comma one have a minimum value at pi comma negative one and end at two pi. But since we are stretching the graph for the first period to end at six pi, that means the ending point for the first period will be at six pi comma one and the minimum value will be located at three pi comma negative one. No notice that we have a negative leading coefficient in front of the cosine function. What this means is that we are going to be rotating the function about the X axis. That means the starting point will be located at zero comma negative one. The ending point will be located at six pi comma negative one and the maximum value will be located at three pi comma positive one. No, we have an amplitude of five. That means that each of the maximum and minimum values of the first period must be five units above and below the X axis. What this means is that the starting point will be located at zero comma negative five. The end point will be located at six pi comma negative five. And the maximum value will be located at three pi com positive five. One more thing to note is that we have a vertical shift of two. What this means is that we are shifting every point on the first period up to units. So the new starting point will be located at zero comma negative three. The ending point will be located at six pi comma negative three. And the new maximum value will be located at three pi comma positive seven. Now, if we were to connect the points together, this is going to give us the shape of the first period of the cosine function. Now, what we need to do is we need to graph the second half of the cosine function. Now the second half of the cosine function is going to mirror the first half of the cosine function. So what this means is that the starting point of the second period is going to be at six pi comma negative three. The ending point is going to be located at 12 pi comma negative three. And the maximum value of the second period will be located at nine pi comma positive seven. And if we were to connect the dots together, this will be the shape of the second period of the cosine function. And with that being said, this is the entirety of the graph of the given trigonometric function with respect to two periods So I hope this video helps you in understanding how to graph this type of problem. And I'll go ahead and see you all in the next video.

Graph each function over a two-period interval.
y = 1 - 2 cos ((1/2)x)

Key Concepts

Amplitude and Vertical Shift of Trigonometric Functions

Period of a Cosine Function

Graphing Trigonometric Functions

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