Graph each function over a two-period interval. See Example 4....

Question

Graph each function over a two-period interval. See Example 4.

y = -1 - 2 cos 5x

Accepted Answer

Hello. Today we are going to be graphing two periods of the given trigonometric function. We are asked to graph Y is equal to negative three minus four cosine of three X. Before we graph the function, there are four pieces of information that we need. We need the graphs amplitude, the period of the function, the phase shift and the vertical shift. In order to obtain these pieces of information, we will be comparing the given trigonometric function to the general function Y is equal to a multiplied by cosine of B multiplied by X minus C plus D. Now that we're gonna use this function, let's start by identifying the amplitude of the given function. Now the amplitude is defined as the absolute value of A, this is where A is the coefficient in front of the cosine function. Now, before we take a look at our given function, let's go ahead and reorder the terms to work. The cosine function is the leading term doing so will allow us to rewrite the function as Y is equal to negative four cosine of three X minus three. The coefficient in front of the cosine function is negative four this means that the amplitude is equal to the absolute value of negative four, which is equal to four. What this means is that the maximum and minimum values will be located at positive four and negative four before any vertical or horizontal shifts. Next, let's go ahead and find the period of the function. Now, the period of the function specifically for cosine is defined as two pi divided by the absolute value of B. This is where the value of B is the coefficient in front of the X minus C turn taking a look at the given given graph, the coefficient in front of the X minus C term is three. What this means is that the period is equal to two pi divided by the absolute value of three which will simplify to be two pi divided by three. What this means is that we are shrinking the period of the cosine function from two pi to two pi divided by three or in other words, the first period of the function will range from 0 to 2 pi divided by three. Now that we have the period, let's find the phase shift of the function. Now the phase shift can be thought of as a horizontal shift of the function, the phase shift will equal to the C value of the general function. Taking a look at our given function, we don't have ac value that we are adding to the X term. So the C value is going to equal to zero for our given function. Or in other words, there is no phase shift to the function. Finally, we need to find the vertical shift. Now, the vertical shift is defined by the D term which is the value we are adding or subtracting to the cosine function. In our given graph, we are subtracting three from the cosine function. What this means is that the vertical shift will equal to negative three, which means that we are shifting the entirety of the graph three units down. Now that we have the pieces of information we need to graph the function. Let's start labeling the general form of the first period. Now we know that the first period of the function is going to end at two pi divided by three. Before we apply any changes to the amplitude or any vertical shifts. The first period of the cosine function will start at zero comma one. We have zeros at pi divided by three. I'm sorry, a minimum value at pi divided by three comma one and end at two pi divided by 31. Now 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 entirety of the first period with respect to the X axis. So that's going to change the starting point to be at zero comma negative one, the ending point to be at two pi comma negative one and the maximum value will be located at pi divided by three comma one. Next, let's go ahead and apply our vertical shift. The vertical shift is going to allow us to move the entirety of the graph down three units. Doing so will give us a new starting point, add zero comma negative four. The ending point at two pi comma 32 pi divided by three comma negative four and a maximum value at pi divided by three comma negative one. Finally, the amplitude is going to be four, which means that the graph is going to stretch four units above and below the maximum values. What this means is that the maximum or the minimum values of the graph will be located at zero comma negative seven. The ending point will be at two pi comma three comma negative seven and the maximum value will be located at pi divided by three comma one. And if we connect the points together, this is going to be the first period of the function. Now we just need to draw the second period with respect to the given changes. The second period is going to mirror that of the first period. So we're going to have the starting point of the second period at two pi divided by three comma negative seven. The maximum value will be at pi comma positive one and the ending point of the second period will be located at four pi divided by three comma negative seven. This will be the second period for a given cosine function and this will be the entirety of the given function. So I hope this video helps you in understanding how to graph this type of function. And I'll go ahead and see you all in the next video.

Graph each function over a two-period interval. See Example 4.
y = -1 - 2 cos 5x

Key Concepts

Cosine Function

Transformations of Functions

Graphing Trigonometric Functions

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