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

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

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Graph the function f (x) = sin 2x + cos 3x.

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Hello. In this video, we are going to be graphing the first rotation of the given sign function, or the given trigonometric function. Now, we are given that F of X is equal to sin of 4 X plus cosine of 4 X. Now, the first thing we need to do is we need to simplify this function into a singular trigonometric function. Recall that we have a trigonometric identity. Where sine theta plus cosine of theta is equal to the square root of 2 multiplied by sine of pi plus pi divided by theta plus pi divided by 4. So, in order to combine these two functions together, we are going to use this identity, but we're going to use the identity where theta is equal to 4 X. So, by applying the identity to the original function, we can condense the function to be F of X is equal to the square root of 2 multiplied by sine of theta which is 4 X plus pi divided by 4. Now what we can go ahead and do is we can reference the general sign function in order to start graphing the trigonometric function. The general sign function is of the form A multiplied by sign, or B multiplied by X minus C. Plus D So the first thing we need to do for our original function is we need to factor the we need to factor out a value from the angle to get B, and we need to switch the innermost angle to be a difference instead of a sum. So what we can go ahead and do is we can factor out of 4 from the angle, and that will leave us with the square root of 2, multiplied by sign of 4 multiplied by X plus pi divided by 16. And if we switch this to a difference, we can state that we have X minus the negative value of pi divided by 16. So, one more way we can rewrite the function is that F of X is equal to the square root of 2 multiplied by sine of 4 multiplied by X minus the square negative pi divided by 16, and that is going to be the general form of the sine function. Now what we want to do is we want to go ahead and identify 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 square root of 2. B is going to be the coefficient in front of the angle, which is 4. C is going to be the value that's being subtracted from X, which will be negative pi divided by 16, and D is going to be the extra value that we are adding or subtracting to the sine function, and that is going to be zero. So, let's go ahead and now first identify our midline of the function. Now the midline is defined by the value D. And since D is equal to 0, that means that our function is going to be centered around the horizontal line Y is equal to 0. So that is just the X axis. Let's go ahead and draw this line. Now what we are going to do is we are going to determine. The amplitude of the function. Now, the amplitude is defined as the variable A. Since we are working with the sine function, the standard, the standard function, or the amplitude is going to be the absolute value of A, which is going to be the absolute value of the square root of 2. But this is going to give us positive and negative square root of 2, which can also be estimated. To be positive and negative, 1.41. So what this means is that we are going, the maximum value that the function is going to reach is 1.41 above and below the midline. We can represent these values by drawing two horizontal lines that are a distance above and below the midline. Now, since we are dealing with 1.41, this is just going to be slightly below 1.5. So these are going to be the lines that represent the maximum and minimum values that our function is going to reach. OK, now that we have the amplitude, let's go ahead and determine the period. Now, the period is defined by the variable B. The standard period of the sine function is 2 pi, so the period in this case will be defined as 2 pi divided by B. Since B is equal to 4, we have 2 pi divided by 4, which simplifies to pi divided by 2. And so what this means is that one full rotation of the sine function is going to have a distance of pi divided by 2. And finally, let's go ahead and determine the phase shift. Now the phase shift is determined by the variable c. Since C is equal to negative pi divided by 16, what this means is that we are going to shift the entire function, a distance of pi divided by 16 to the left. So, let's go ahead and start by determining the endpoints of the function. Now the standard sign function. Has an S-like shape. It starts at the origin. And ends at the midline on the x-axis. Now, in this case here, because our period is pi divided by 2, before applying the phase shift, our endpoints are going to be 0 to pi divided by 2. But we are going to subtract pi divided by 16 to both of the end points. So that is going to be 0 minus pi divided by 16, and pi divided by 2 minus pi divided by 16. Now, this is going to give us negative pi divided by 16 as our first endpoint, and we are going to have 7 pi divided by 16 as our final endpoint. So, let's just go ahead and label these endpoints on the midline. So that is going to be the endpoints of our of our graph. Now, what we want to do is we want to determine the behaviors of the rest of the graph. Recall that the standard angles we work with on the unit circle are 0, pi divided by 2, i, 3 pi divided by 2, and 2 pi. So what we want to do is we want to determine the angles for pi divided by 2, pi and 3 pi divided by 2. In order to do this, we can take the innermost angle of the function, 4 X plus pi divided by 4, set them equal to the remaining +33 angles, and solve for the remaining values of X. So For the 1st 0.4 X plus pi divided by 4 equals pi divided by 2. Now, this is going to be the point at the next maximum value of the function. Solving for X is going to give us X is equal to pi divided by 16, and this is going to have a Y value of square root of 2. So this will be our first point. Next, we will take 4 X. Plus pi divided by 4. And set this equal to pi. Solving for X gives us X is equal to 3 pi divided by 16, and since the midpoint of the sine curve, this is going to have a Y value of 0, which will be our next point. And finally, We will take 4 X plus pi divided by 4, and set this equal to 3 pi divided by 2. This is going to give us an X value of X is equal to 5 pi divided by 16, and since this is the point at the minimum value of the graph, this will have a Y value of negative square root of 2. Now let's go ahead and plot the remaining points. So by estimating This is going to be the rough shape. Of the rest of the graph. And this will be one full rotation of the given sign function. So I hope this video helps you in understanding on how to approach this problem, and 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) = sin 2x + cos 3x.

Key Concepts

Trigonometric Functions

Graphing Functions

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