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 = -2 cos 3x

Accepted Answer

Hello, today we are going to be drawing the function given below, we will be drawing two periods of the function and we will be determining the period and amplitude we are given Y is equal to negative four multiplied by cosine of two X. So before we graph this function, let's go ahead and identify the period and amplitude. In order to identify the period in amplitude, we can compare the given graph or given equation to the general equation. 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 and this is where the A term is the coefficient in front of the cosine function in our given equation. This coefficient is negative four. What this means is that the amplitude of the equation will equal to the absolute value of negative four. And this was simplified to leave us with positive four. What this means is that the amplitude is going to be four units above and below the X axis. Or in other words, the range of the function will be from negative four to positive four. So now we will identify the period or the function. The period for cosine is defined as two pi divided by the absolute value of B. This is where the B term is the coefficient in front of the X minus C term. And that coefficient is given to us as positive too. That means our period is going to equal to two pi divided by the absolute value of two. This will simplify to leave us with two pi divided by two and will further simplify to just be pi. So what this means is that the first period of our function will start at zero and end at pi. Now that we have this information, let's go ahead and graph the first period of the function. Now the starting point for cosine is traditionally at zero comma one. But since our amplitude is four, the new starting point for our cosine function will be at zero comma positive four. Now, since our period ends at pi, the ending point for our cosine function will be at pi comma positive four cosine will have two zeros located at pi divided by four and three pi divided by four. And the minimum value will be located. But at pi divided by two comma negative four, this will be the first period of our given cosine function. But there is one transformation that we need to keep in mind cosine has a negative leading coefficient. And that means we're going to need to rotate every point of our period about the x axis. That means the new starting point is going to be located at zero comma negative four. The ending point will be located at pi comma negative four. And the maximum value will be located at pi divided by two comma positive four. Once we connect the dots together, this is going to be the shape of the first period of our cosine function. Now, what we need to do is we need to graph the second period of our cosine function. Now, since the period is pi, we can identify the ending point by adding another instance of pi to pi pi plus pi will give us two pi. And that means that the second period of our cosine function will end at two pi the minimum or the maximum value of our cosine function will be located halfway between pi and two pi which is three pi divided by two. And this maximum value is four units above the X axis. So the, so the maximum value of the second period will be located at three pi divided by two comma positive four, the zeros will be located at five pi divided by four and seven pi divided by four. And once we connect the dots together, we will have the second period of our cosine function. And this will be the entirety of the graph of two periods of a given trigonometric equation. 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 = -2 cos 3x

Key Concepts

Period of a Trigonometric Function

Amplitude of a Trigonometric Function

Graphing Trigonometric Functions

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