Use graphing software to graph the functions specified in Exer...

Question

Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.

Graph the function f (x) = sin³ x.

Accepted Answer

Hello, in this video, we are going to be drawing one period of the given sine function. So we are given that sin F of X is equal to sin of 2 X minus pi divided by 3, raised to the 3 power. Now, in order to approach this problem, let's just quickly recall what the general form of the sine function is. The general form of the sine function is given to us as F of X is equal to A multiplied by sine of B multiplied. By x minus c. Plus D. So, in order to approach this problem, we need to first rewrite the inside side function in the form of the general sign function. And in order to do this, we will need to factor out a 2 from the given sign function. So we're going to focus on the innermost sign function for now, and that is going to be F of X's equal to sine. Of 2 multiplied by X minus pi divided by 6 cubed. Now, focusing on the innermost sign function, let's go ahead and first start by identifying our our coefficients A, B, C, and D. Now, A is the coefficient in front of the sine function, which in this case is going to be one. B is the coefficient in front of the angle, which is 2. C is the value that X is being subtracted by, which is going to be pi divided by 6, and D is the extra value that we were adding or subtracting to the graph, but in this case, this is going to be 0. Now, let's go ahead and start by talking about the midline of the function. Now, the midline is defined by D and the midline is going to be the value or the horizontal line that the function is centered around. In this case here, D is equal to 0, so our midline of the function is just going to be the X axis. Defined by Y is equal to 0. Now let's go ahead and figure out the amplitude of the function. Now, the amplitude is defined by the variable A. Since A is equal to 1 in this case, the amplitude will be defined by the absolute value of A, which is the absolute value of 1, and this is going to give us positive and negative 1. What this means is that the amplitude or the maximum and minimum values of that the function will reach, will be one unit above and below the midline. So we can represent these values by drawing horizontal lines. So the graph is never going to go above or below the two black lines that we just drew. Now let's go ahead and talk about the period of the function. Now the period is going to be defined by the variable. B Now, the standard period for the sine function is going to be 2 pi. So the period will be defined as 2 pi divided by B and since B is equal to 2, we will have 2 pi divided by 2, which gives us the value of pi. What this means is that one full rotation of the sine function is going to start at the origin but end at pi. So we can draw that stopping point with this vertical line. OK. And finally, let's go ahead and talk about the phase shift. Now the phase shift is defined by the variables C. Since C is equal to pi divided by 6, that means we are going to shift the entirety of the function over by 1 value of pi divided by 6. So in the meantime, let's just go ahead and clear up this vertical line we drew. But now what we want to do is we want to go ahead and determine what the endpoints of the first rotation is going to be. So, the standard sign function starts at the origin. And ends at 2 pi. But in this case here, since we have a period of pi, that means that the starting point of the original sine function will be zero, but the ending value will be pi. But we are going to take these end points and shift them over pi divided by 6 units, since the phase shift of the entire function is pi divided by 6. So that means that we're going to add the value of pi divided by 6 to both endpoints. 0 plus pi divided by 6 will give us pi divided by 6, and pi plus pi divided by 6 will give us 7 pi divided by 6. So what this means is that the starting point of the graph is going to be located at pi divided by 6 on the midline, and it's going to end at 7 pi divided by 6 on the midline. Now we just need to determine the behaviors in between the graph. So, recall that the standard angles that we work with on the unit circle are 0, pi divided by 23 pi divided by 2, and 2 pi. What we want to do is you want to determine what the angles are at pi over 2, pi and 3 pi over 2. So, we are going to take the original angle given to us, 2 X minus pi divided by 3, and determine what the points are going to be at those values. So we're taking 2 X minus pi divided by 2. And we're going to set this equal to the first angle we are interested in, which is pi divided by 2. Solving for X is going to give us the value of 5 pi divided by 12, and since it is the first behavior or the first point after the origin, it is going to be located at the maximum value of the graph, which is going to be positive 1. So that will be our first point. Now, the next point is going to be located on the midline at 0. We're going to take 2 X minus pi divided by 2 and set this equal to pi. Solving for X is going to give us 2 pi divided by 3, and the Y value for this point will be 0. And for the final point, we will take 2 x minus pi divided by 2, set this equal to 3 pi divided by 2, and solve. This is going to give us the angle X is equal to 11 pi divided by 12 with a y value of -1, since this point will be located at the minimum value of the sine curve. So now that we know what the next three points are going to be, let's go ahead and plot them. Now, 2 pi divided by 3 is going to be located on the midline, so the third point will be located here. Now The first, the second point of the function, 5 pi divided by 12, is going to be located some distance between pi divided by 6 and 2 divided by 3. So we can estimate that distance to be here at the maximum value, and the same assumption can be made with the fourth point of the function. And if we connect the points, this is going to be the graph of the first rotation of the original sign function. So I hope this video helps you in understanding on how to approach this problem, and then we will go ahead and see you all in the next video.

Use graphing software to graph the functions specified in Exercises 31–36.
Select a viewing window that reveals the key features of the function.

Graph the function f (x) = sin³ x.

Key Concepts

Graphing Functions

Key Features of Functions

Viewing Windows in Graphing

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