Graph each function over a two-period interval. Give the perio...

Question

Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.

y = ½ cos π x

2

Accepted Answer

Hello, today we are going to be graphing two periods of the given trigonometric function and we will be determining the period and amplitude of the function. So we are given Y is equal to one divided by four cosine of pi divided by three X. Before we jump into this equation, let's go ahead and identify the period and the amplitude, we can obtain the period and the amplitude by comparing the given trigonometric function to the general function Y is equal to a cosine of B multiplied by X minus C plus D. The amplitude will equal to the absolute value of A. This is where A is the leading coefficient in front of the cosine function that coefficient is given to us as one divided by four. What this means is that our amplitude is going equal to the absolute value of one divided by four. This will simplify to be positive, one divided by four and can be rewritten to be 0.25. What this means is that the amplitude of our cosine function is equal to 0.25. The maximum and minimum values will be located 0.25 units above and below the X axis. And that means our range is going to equal from negative 0.25 to positive 0.25. Now that we have the amplitude, let's go ahead and identify the period. The period of a cosine function can be ident or written 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. This coefficient is given to us as pi divided by three. What this means is that the period is equal to two pi divided by the absolute value of pi divided by three. This will simplify to leave us with two pi divided by pi divided by three and will further simplify to give us a period of six. What this means is that the first period of the cosine function will be from 0 to 6 now that we have the period and we have the amplitude, let's go ahead and start graphing the cosine function. Now typically the starting point of the cosine function will be located at zero comma one. But since our amplitude is 0.25 that means our starting point of the cosine function will be located at zero, comma 0.25. Since our first period ends at six, the ending point of the period will be located at six comma zero, the minimum value will be located halfway between zero and six on the X axis which is going to be three. And since our minimum value is located 0.25 units below the X axis, the minimum value of the cosine function will be located at three comma negative 0.25. Now the zeros of the cosine function will be located halfway between zero and three and halfway between three and six. This value is going to be 1.5 which is roughly located at these spots right here. Once we connect the dots together, this is going to be the rough shape of the first period of our cosine function. Now that we have the first period of our cosine function, we need to graph the second period of our cosine function. The second period will have the same shape as the first period. Since our period has a length of six units on the X axis, that means the ending point of our second period will be located at 12. The minimum value of our second period will be located halfway between six and 12, which is going to be nine and this minimum value will be negative 0.25 units below the X axis. So the minimum value of our second period is located at nine comma negative 0.25. The zeros of the graph will be located halfway between six and nine and halfway between nine and 12. These values are going to be roughly at 7.5 and at 11.5. And once we connect the dots together, this will be the second period of our given trigonometric equation. And in total, this is going to be the shape of two periods of our given cosine function with the amplitude of 0.25 and period of six. 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 a two-period interval. Give the period and amplitude. See Examples 2–5.
y = ½ cos π x
2

Key Concepts

Period of a Trigonometric Function

Amplitude of a Trigonometric Function

Graphing Trigonometric Functions

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