Beginning with the graphs of

Question

Beginning with the graphs of $y=\sin x$ or $y=\cos x$ , use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work.

$q\left(x\right)=3.6\cos\left(\frac{\pi x}{24}\right)+2$

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to draw the graph of the following trigonometric function using transformations from the basic trigonometric function, and we're given the function F of X is equal to 2 multiplied by the cosine of the quantity of pi multiplied by X divided by 18 in quantity plus 3. Now, we need to create a graph of this function, using transformations from our basic trigonometric function, which in this case is going to be cosine X. So, we're gonna be looking at several different properties, and the first one is going to be the period. And so recall for the period of our basic trigonometric function, which is going to be cosine X. That it has a period of 2 pie. And so we want to find the period of the function of cosine of the quantity of pi multiplied by X divided by 18. And so to find its period, we're going to take our original period, which is to pi from our basic function, and divide it by the coefficient of X, which in this case is pi divided by 18. And so when we simplify this fraction, the pies will cancel and we get 2 multiplied by 18, which is equal to 36. So basically what we've done is stretched out our cosine curve by a factor of 18 divided by pi along the X direction. Now, the next quantity that we want to look at is the amplitude. And for the amplitude, that's going to tell us how far away that it has oscillated from our midline. And so we call for the cosine of X, its amplitude is equal to 1. And here, we're going to be looking at the function of 2 multiplied by cosine of the quantity of pi X divided by 18. And so our amplitude is going to be the coefficient in front of our cosine function, which in the case of cosine X, which was just one since one is multiplying cosine X. But here in our new function for F of X, we have cosine B multiplied by 2. That means our amplitude or 2 cosine of pi X divided by 18, its amplitude is going to be 2. So, I mentioned just a minute ago, something about a midline, so now we need to look at where our midline is located. And so this is the line. That our um function is oscillating about. And for cosine of X. The midline that it is oscillating about is the line Y equal to 0. So it is oscillating about the X axis. Now, we're going to be looking at the function of 2 multiplied by cosine of the quantity of pi X divided by 18. Plus 3. So, the plus 3 is where our midline is going to be located. So it's going to be located at Y equal to 3, since we've shifted up our entire graph by 3 units. So the last quantity that we need to calculate is going to be our range. And the range for cosine effects, if you recall. is between -1 and 1. And now if we use that information to look at our range for cosine, the two cosine of pi X divided by 18. Plus 3. Then this is going to oscillate about Our midline, which is 3, plus or minus our amplitude. So for the lower bound we'll have 3 minus 2, and for the upper bound we'll have 3 + 2. So that means that Our function is oscillating between the values of 1 and 5 on the Y axis. So now we have all the information that we need to to actually draw our graph. And so we're going to be looking at plotting some points on our grid. And the first thing we need to recall is that when X is equal to 0 or our cosine of X function, it is at a maximum. And the same thing is going to be true for a function F of X. When X is equal to 0, it's going to be at a maximum, which the maximum range in our case is 5. So we'll place a point at 0.5. Now we're going to play some other points on a graph to help us actually create our graph. The first one is going to be the point where our function crosses the midline when Y is equal to 3, and this occurs when we have gone through a quarter of a period. And since our period is 36, if I divide 36 by 4, I get 9. So, our next point is going to be at 9.3. That's what we'll first cross the midline. Then we will have a minimum when we've gone halfway through a period. So, half of 36 is 18. And so, I will plot the point of 18.1, since 1 is our minimum value of our range. Then our function will oscillate back up, passing the midline again. In this case, it's going to occur when X is equal to 27. So we plot point at 27.3. Then we'll hit another maximum once we've completed one full period, and that's going to occur at 36.5. We'll place a point there, and we can cross the midline one more time, when X is equal to 45, we'll place another point at 45.3. Now, we've taken care of everything along the positive X direction, we'll just mirror everything on the negative X direction. So we'll place a point at -9. 3. Another point at -18.1. Another point at -27.3. Another point at -36.5, and our final point at -45.3. So now what we need to do is carefully connect these dots following a cosine curve. And this can be difficult to draw by hand. But here we're just looking for more of a sketch, just trying to connect every single point so we can get a good visualization of what our curve will actually look like. And another thing that you could do is double check your curve using a graphing utility. So, if it becomes a little too difficult to draw your curve, you can always use a graphing utility to check. To make sure that you've done everything properly, and that you haven't um made any mistakes while connecting your dots. And so, we're just taking some extra care, making sure that we connect all the dots, and then what we'll do is we will extend um our Function down to the edges of the graph just by trying to guess where the next point will be. And so what we have here is the final answer to our question. This is what the question was asking for, and this is the graph of our function F of X, which is equal to 2 multiplied by cosine of the quantity of pi X divided by 18 and quantity plus 3. So I hope that this explanation has been useful, and I'll see you in the next video.

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