Graph y = 1/2 sin x + cos x, 0 ≤ x ≤ 2π.

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Hello. Today we're gonna be drawing the graph for the following function on 0 to 2 pi. So what we are given is Y is equal to 1/4 sine X plus cosine X. So how do we go about drawing the graph for the following function? Well, one thing to note is that we want the graph on the domain of 0 to pi this is the definition of one rotation of the unit circle. And so what we can go ahead and do is we can use the major angles of the unit circle to help us find the points for the graph. The unit circle starts off at zero degrees, then it rotates counterclockwise in the positive direction. The major angles of the unit circle are going to be zero pi over two pi three pi over two then ends at two pi. So we can use these five angles as our inputs to the function. And what we want to do is use these inputs to get the outputs for the points of the graph. So let's go ahead and draw a chart. We're going to have 1/4 sin of X. We're going to have cosine of X, then we're going to have the sum of these two values. Then for the columns of the chart, we're going to have the input values for each of these functions. The first input value is going to be zero. If we input zero into sin of X, the output is going to be zero. If we input zero into cosine of X, we get the output of one and adding zero and one together is going to give us the value of one. So that tells us that the first coordinate for the graph is going to be at zero comma one. Next, let's do the next input point which is going to be pi over two. If we plug in pi over two into 1/4 sine X, we get the value of 0.25. If we plug in pi over two and a cosine X, we get zero and the sum of T these two values is going to be 0.25. So the next coordinate on the graph of the function is going to be at pi over two comma 0.25. The next input is going to be pie inputting pi into 1/4 sine X is going to give us the value of zero in putting pi into cosine is going to give us the value of negative one and zero minus one is going to give us negative one. So the next coordinate is going to be at nip at pi come a negative one. The next input will be three pi over two inputting three pi over two into 1/4 sin sin gives us the value of negative 0.25 and cosine of three pi over two is going to give us the value of zero. So the sum of these two values is going to give us negative 0.25. And the next coordinate is going to be three pi over two comma negative 0.25. And the final input for the function is going to be two pi 1/4 sign of two pi will give us the value of zero cosine of two pi will give us the value of one and zero plus one is going to give us one. So the final coordinate for the graph of the function is going to be two pi comma one. So now that we have five coordinates that we can plot for the graph. Let's go ahead and draw an axis for the graph. So we have an axis and the domain of the graph is going to be from zero to two pie. Pie is going to be right in the middle of this graph and in between zero and pi is going to be pi over two and in between pie and two pie is going to be three pie over two. And the amplitude for the graph is going to be between one and negative one. So let's go ahead and start by plotting our first point. We know that the first point of the graph is going to be at zero comma one. The next value is going to be at pi over two comma 0.25. So it's gonna be between zero and one on the Y axis and it's going to have an X value of pi over two. So roughly in this location, the next point is going to be at pi and negative one. So Pi is right down the middle on the X axis and negative one is the very bottom of the amplitude. The next value is at three pi over two and negative 0.25. So that location will be roughly located here and the final point is going to be a two pi comma one. So that'll be located at the same height as zero comma one. Now, if we connect two points together, we're going to end up with this bell shaped curve and this is going to be the rough graph for the given 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 y = 1/2 sin x + cos x, 0 ≤ x ≤ 2π.

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