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 = sin 3x

Accepted Answer

Hello, today we are going to be drawing the given function below. We will be drawing two periods of this function and we will be determining the period and amplitude of the function. So we are given Y is equal to sine of five X. Before we start graphing this function, let's go ahead and first identify the period and the amplitude, we can obtain the period and the amplitude of the function by comparing our given function to the general function Y is equal to a sine of B multiplied by X minus C plus D. The amplitude will equal to the absolute value of A and this is where A is going to be the coefficient in front of the sine function. If we take a look at our given functions has a coefficient of one. What this means is that the amplitude is going to equal to the absolute value of one which will simplify to be positive one. Now, the standard sine function has an amplitude of one. So what this means is that our given function has no change to its amplitude and the range of the function is going to remain from negative one. To one. That means that the maximum and minimum values of our sine function are going to be located one unit above and below the X axis. Now let's go ahead and identify the period of the function. The period of a sine function will be 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. This coefficient is given to us as five in front of the X term. So that means our period will be defined as two pi divided by the absolute value of five. And this will simplify to leave us with two pi divided by five. So what this means is that one period of our function is going to be from 0 to 2 pi divided by five. Now, the sine function always starts at the origin since our period is going to be two pi divided by five. That means the first period of our sine function will be ending at two pi divided by five. The zero of our sine function for the first period will be located at the halfway point between zero and two pi divided by five. This zero is going to be located at pi divided by six. Our maximum value will be found at the halfway mark between zero and pi divided by six. This will be pi divided by 10 and our maximum value will be located one unit above the X axis. So our maximum value will be pi divided by 10 comma one. Our minimum value will be located halfway between pi divided by six and two pi divided by five. And since the amplitude is one, it'll be located one unit below the X axis. So the minimum value for the first period will be located at three pi divided by 10 comma negative one. Once we connect the dots together, we will have a snakelike shape. And this will be the first period of our given equation. Now we need to graph the second period of the equation recall that the period is two pi divided by five. In order to identify the ending point of our second period, we are just going to add two pi divided by 5 to 2 pi divided by five. This is going to give us an end of four pi divided by five. The zero of our second period will be located halfway between two pi divided by five and four pi divided by five. This will be three pi divided by five. The maximum value for the second period will be located halfway between two pi divided by five and three pi divided by five. This maximum value will be located at pi divided by two and one unit above the X axis. And our minimum value will be halfway between three pi divided by five and four pi divided by five. This will be located at seven pi divided by 10 and will be one unit below the X axis. Once we connect the dots together, this will be the second period of our function. And in total, this is going to be two periods of our given trigonometric function. 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.

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Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.
y = sin 3x