Graph the functions in Exercises 13–22. What is the period of ...

Question

Graph the functions in Exercises 13–22. What is the period of each function?

−cos 2πx

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Graph the function Y equals negative cosine of 4 pi X and determine its period. Consider only 1 period when graphing. Awesome. So it appears for this particular problem we're ultimately trying to create a graph for this function of Y equals negative cosine of 4 pi X. So now that we know that we're trying to graph this function, First off, what we need to do is we need to note that the given function is in the form of Y equals. A multiplied by cosine of B X minus C plus D. Where the absolute value of A, where A is the amplitude. And 2 pi divided by B is equal to the period, and C divided by B is the phase shift, and D is the vertical shift or the translation, as we should recall. Fantastic, and note that anything inside of the absolute value symbols will become a positive value. So with that said, the amplitude for this particular problem is going to be equal to -1, so the absolute value of -1 is going to be simply 1, and we're told that our value for B for the period, so we need to take 2 pi divided by B is going to be equal to 2 pi divided by 4 pi because we're told that B in this case is 4 pi, and when we perform the calculation, we will find that it's simply equal to 1/2, and then the phase shift, which is C divided by B, is going to be equal to 0 divided by 4 pi, which is going to be simply equal to 0. And the vertical translation D is going to be equal to 0 because there's no value of D in our particular function that is given to us. So now we need to divide the period into 4 equal parts, so we need to take our period divided by 4. So we need to take our period, which we're told that our period is 1/2, and we need to divide it by 4, which will give us an answer of 1/8. Fantastic. And with that said, we now need to solve for X1, and we're told that X1 is going to be equal to 0, and X2 is gonna be equal to X1 plus 1 divided by 8, which is going to be equal to. 1/8 because we're plugging in a value of 0 in for X1. X3 is going to be equal to the value of X2 + 1 divided by 8, which will give us an answer of 1 divided by 4 or 1/4. And X4, so X subscript 4 is going to be equal to the value of X3. Plus 1 divided by 8, which will give us an answer of 3/8 and X or I should say X to the subscript of 5, so X5, it's going to be equal to the value of X4. Plus 1/8, which is going to be equal to 1/2. Fantastic. And now we need to evaluate our function, which, let's take a moment here to quickly recall that our original function, we're told that that is going to be equal to Y is equal to negative. Cosine of 4 pi X. So we're gonna be plugging in our different values of X that we found into our original function to solve. So we're gonna take Y1 is equal to Y of X1, which is going to be equal to Y of 0, which when we plug in a value of 0 in, we will find that this is going to be equal to negative cosine of 4 pi multiplied by 0. Which will give us an answer of simply -1, and instead of writing this this whole long expression out every single time, I'm just gonna give you what the end values for For Y2 through 5 is. So, Y2, Y3, Y4 and Y5. So Y2 is going to be equal to 0. Y3 is gonna be equal to 1. Y4 is going to be equal to 0, and Y5 is going to be equal to -1. And once again, we're just plugging in our different values for X2, X3, X4, and X5 into our original function. And when we do, we will receive the following answers. So now we need to plot the points. Parentheses 0.1, you need to plot 180 in parentheses. So we're trying to plot these five points, so We got 2 points, so now we need to do our third point, which is 1/4 1. And 3/8.0, so that's 4, and then our last point that we're gonna need to plot is 12-1. So we got 1234, and 5 points that we need to plot on our graph. So when we plot these 5 points onto our graph, And we connect them with a smooth curve, we should produce the following graph. So as you can see, looking at our five points which are represented in yellow, we have our point at 0.1 and at 1/8.0 at 14.1, at 3/8.0, and at 12 1. So it starts touching the Y axis, and then it starts to curve upwards where it crosses through the X-axis till it peaks, and then it starts to curve down till it crosses the X axis at its second point, and then dips down to our last coordinate point, which is at 1/2 1. And our curve is represented in blue, and that's it. We've officially solved for this problem. Hooray, we did it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Graph the functions in Exercises 13–22. What is the period of each function?

−cos 2πx

Key Concepts

Period of a Function

Cosine Function

Graphing Trigonometric Functions

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