WEBVTT
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Let's explore the graph of the sine function, y equals sine x. Here I've made a table where x is the angle
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in standard position and y is the value of the sine function. Notice that all the angles in my table are special angles,
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so you should be able to determine all these values on your own. Then I've plotted all the ordered pairs,
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as you can see, so now let's see what the graph of sine x looks like. Let's connect our points.
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So we have a nice graph of the sine function. If I had chosen values bigger than 2 pi, say 2 pi plus pi over 6
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that would be 13 pi over 6. The value of the sine function of 13 pi over 6 would be what?
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One half, exactly. So you can see that this pattern is going to repeat itself, if I had chosen values bigger than 2 pi
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or less than zero. Now let's look at some characteristics of the sine function now that we know the graph.
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Here is the graph of y equals sine x. Let's look at some of the characteristics of the graph. The first thing we want to
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notice is what's the domain? Remember domain means the values of x. Well notice that the left side there's arrows,
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so it's going to negative infinity on the right, on the left, excuse me. And on the right it's going to positive infinity.
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So the domain would be all real numbers. Now let's look at the range of the function, the sine function.
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That's looking at the y-values. What do you notice about the y-values? Notice the smallest y-value is negative 1,
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the largest y-value is 1, and it takes on all values in between. So the range would be negative 1 to 1.
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Another thing we notice about this, there are three periods shown here. Here's one period, here's another period,
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and another period where the sine function is repeating itself three times. So that means the period of these intervals
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in between here, here's another period, and finally we have one more period drawn here. So the period for the sine
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function would be 2 pi. Now what that means is that if you're trying to find values, that means the sine of some value
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x is going to be the same thing as the sine of x plus 2 pi times, say n, where n is any integer. They'll be the same value
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because the sine function is periodic. Now, another thing we notice about the graph is that sine is an odd function.
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What does that mean? That means that the sine of negative x is the opposite of the sine of x. That also means
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the graph is symmetric about the origin. Another characteristic, let's look at the intercepts. What's the y-intercept?
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Where does the graph cross the y-axis? We can see that the y-intercept is 0. Now, what about the x-intercepts?
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Well, there's an infinite number of them, as we can see, because you remember that this goes on forever, but let's
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describe where they occur. Well, let's see. One x-intercept is at zero, and then pi, and then 2 pi, and so on.
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Do you see the pattern? So the x-intercepts are at all multiples of pi. So we can describe that by just saying n pi,
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again where n is an integer. Two more things we want to notice about the sine function. Notice that there's a maximum
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point and a minimum point, a relative max and a relative min, and we want to talk about where they occur.
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Well we know there's an infinite number of them because the sine function repeats itself. Let's look at the relative
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max and talk about where they occur. The relative max occurs at what x-values? Well, let's see. The first one occurs,
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well not the first one, but one of them occurs at pi over 2. Where is the next one? Well, it's going to be a period away,
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2 pi away. So I can describe where they occur by saying at x equals pi over 2 plus 2 pi n, where n is any integer.
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And then of course now we see there's a relative min. Where does it occur? The relative min occurs at what x-value?
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Well one of them is 3 pi over 2, and again to get to the next one we add multiples of the period, so we can say at
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3 pi over 2 plus 2 pi n, again where n is an integer. You should learn all these characteristics of the sine function.
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They're very important. Now we're going to focus on just one period of the sine function, and look at some of those
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important points that you're going to need to know.
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Here is the graph of y equals sine x with x between 0 and 2 pi or one period or cycle of the sine function,
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and we want to notice some important points on this cycle. Those important points occur at the x-intercepts
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and the y-intercept, which of course is the same in this case. So I've got the x-intercepts and then the points where
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the relative max and the relative min occur. Notice that these points occur at one fourth of the period, so we call them
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quarter points. Let's list the quarter points. The first one is the origin, (0,0). The next one is (pi over 2,1).
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Then (pi,0), (3 pi over 2,-1) and (2 pi,0). These are important points for you to know, so you should know these
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quarter points for the sine function so that you can graph it easily.
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Let's explore the graph of the cosine function, y equals cosine x. Here I've made a table of values where x is the
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angle in standard position in radians and y is the value of the cosine function. Notice that all the angles in my table
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are special angles, so you should be able to get all these values on your own. Then I plotted the ordered pairs,
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as you can see here, so we can see what the graph of the cosine looks like. Let's connect our points. So let's see.
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It's going to start up at 1, go down, and do this. So I've plotted the points from 0 to 2 pi. What would happen
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if I chose values bigger than 2 pi, or say, less than zero? Let's take an angle. For example, 2 pi plus pi over 6
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would be 13 pi over 6. What would the cosine of 13 pi over 6 be? Well let's see. That's in quadrant one, cosine's
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positive, so it's going to be the square root of 3 over 2. So we can see that this pattern will just repeat itself
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for values bigger than 2 pi or less than zero. Now that we know what the graph of the cosine looks like,
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let's look at some characteristics of the graph.
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Let's look at the graph of the cosine function. Here I've got three periods of the cosine function, and we want to look
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at some of the characteristics. Well the domain of the cosine function, the values of x, are all real numbers.
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We can see that the graph goes from negative infinity to positive infinity, so the domain is all real numbers.
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The range will be the y-values. Let's look at those. The y-values go from negative 1 up to 1 and take in all values
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in between, so the range would be negative 1 to 1. The period is 2 pi. We can see here's a period, between here,
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that it repeats itself again here, so here's another period of 2 pi, and then I see another period over here.
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So the period for the cosine function is 2 pi. Now let's look at another characteristic which would be I notice that
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this function is symmetric about the y-axis which means it's an even function. That means that the cosine of
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negative x is the same as the cosine of x. It's symmetric about the y-axis. Next let's look at the intercepts. Well,
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the y-intercept is what? The y-intercept is 1. It crosses the y-axis at 1. What about the x-intercepts? Let's look at those.
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Where do they occur? Well we can see we have infinitely many of them. We can see one, two, three, four, five, six
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here on these cycles here. But where do they occur? Well, let's see. pi over 2, then the next one's at 3 pi over 2,
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5 pi over 2, 7 pi over 2. Can you predict where the next one would be?
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9 pi over 2, exactly. So you can see that they occur at odd multiples of pi over 2. How do we write that?
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Well, odd number is any number 2n plus 1 gives you an odd number, and we multiply that odd number times
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pi over 2. That's where all the x-intercepts occur. The other thing we want to look at is the graph has relative max
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and relative min points, and we want to talk about where they occur. Let's look at the relative max first. Where does
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it occur? Well this one's of course at x equals 0, and then the next one occurs 2 pi away so we can add multiples
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of the period 2 pi, which is the same thing of course as saying 2 pi n where n is any integer. Now let's look at the
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relative min. Here's a relative min. Where does it occur? At pi. Where does the next one occur? A period away, this
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is an interval of 2 pi, so the relative min occurs at x equals, this would be pi plus 2 pi n again where n is an integer.
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So here are the characteristics of the cosine function that you should be very familiar with.
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Now I want to focus on just this one period and look at some important points.
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Here we have one cycle of the cosine curve, and we want to look at some important points. The important points
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are where the intercepts or the relative max, the x-intercepts, and the relative min. Those points occur at one fourth
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of the period, so we call them quarter points. Let's list the quarter points for the cosine function between 0 and 2 pi.
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Well the first one is at 0, (0,1). The next one occurs at (pi over 2, 0). The next point here is (pi, -1), (3 pi over 2, 0),
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and finally (2 pi, 1). So here are the five quarter points for cosine x between 0 and 2 pi. You need to know
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these quarter points for the cosine function so you can graph it.
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Here we want to graph functions of the form y equals A sine x and y equals A cosine x to see what effect A has
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on the graph. First thing we need to talk about is the amplitude of these graphs will be the absolute value of A.
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It's always positive. Let's look at an example. So I have this function y equals 3 sine x. What's going to happen is
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this 3 is going to be multiplied by all these y-coordinates of the quarter points. So you can see that I've got all
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those quarter points of sine x listed. So now let's figure out what the quarter points will be for 3 sine x.
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Well it doesn't change the x-coordinates, so they're still going to be 0, pi over 2, pi, 3 pi over 2, and 2 pi.
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So to get the y-coordinates for 3 sine x we're going to multiply these values by 3. So 0 times 3 is of course 0.
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1 times 3 is 3. 0 times 3 is 0. Negative 1 times 3, negative 3, and then again 0 times 3 is 0. So here will be
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the quarter points for y equals 3 sine x.
Let's graph that function.
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So we have (0, 0), then we have (pi over 2, 3), (pi , 0), (3 pi over 2, -3), and (2 pi, back up to 0). So the amplitude is
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this distance, half the distance between the max and the min value. Let's write the amplitude for y equals 3 sine x.
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The amplitude will be the absolute value of 3 which is 3. Again, that's the distance, half the distance between the
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relative max and the relative min, which makes the range of this graph what? Exactly.
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The range is now going to be negative 3 to 3. Now let's look at another example.
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Let's look at the graph of y equals negative 2 cosine x. The amplitude will be the absolute value of negative 2,
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which of course is positive 2. Now, in order to graph this function, what we're going to do is multiply negative 2
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by our y-coordinates of our quarter points for y equals cosine x. So let's write the quarter points for y equals
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negative 2 cosine x. Well the A only affects the y-coordinates so the x-coordinates will not change.
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Ok, so let's look at the first one. So the y-coordinate of cosine x is 1, multiply by negative 2, and what do you get?
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Negative 2. It's important to note that you don't multiply your y-coordinates by the amplitude. You multiply your
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y-coordinates by A. Alright so let's keep going. 0 times negative 2 is still 0. Negative 1 times negative 2 is positive 2.
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0 times negative 2 is 0, and 1 times negative 2 is negative 2. So here are our quarter points
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for y equals negative 2 cosine x. Let's graph this function.
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So let's see. I have (0, -2), and (pi over 2, 0), (pi , 2), (3 pi over 2, 0), (2 pi , -2). So here's the graph
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of y equals negative 2 cosine x. What is the range for this graph? Well the range is going to be
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negative 2 to positive 2. So in general when you're graphing functions y equals A sine x or y equals A cosine x,
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the amplitude will always be the absolute value of A,
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and the range will always be negative absolute value of A to positive [absolute value of] A.
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Here we want to look at graphing functions of the form y equals sine Bx and y equals cosine Bx. For these functions,
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the period will be 2 pi divided by B. Now it's important to note that B must be positive here. Let's look at an example.
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y equals cosine 3x, so B is positive 3. So the period's going to be 2 pi divided by 3. So B is going to change
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the x-coordinates. Here I've listed the quarter points for y equals cosine x which of course has a period of 2 pi.
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For cosine 3x, the period is 2 pi over 3. The quarter points occur at one fourth of the period, so let's figure out,
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what is one fourth of this period? One fourth of 2 pi over 3 is pi over 6, so now our x-coordinates are going to be
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in increments of pi over 6. The y-coordinates will stay the same. So let's list the quarter points for y equals cosine 3x.
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Ok, so the first one still occurs at 0. The next quarter point is going to be pi over 6 increment, so you're always
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adding pi over 6 to each x-coordinate to get the next one. What is pi over 6 plus pi over 6? Well, it's 2 pi over 6,
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but of course you always reduce your answers. Ok, and then 2 pi over 6, or pi over 3, plus pi over 6 gives you
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3 pi over 6 which is pi over 2. And finally, 3 pi over 6 plus another pi over 6 is 4 pi over 6 or the period of 2 pi over 3.
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So here are the quarter points for y equals cosine 3x. Let's plot those points. So we have a period of 2 pi over 3.
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And let's see. Notice the y-coordinates are going to be 1 and negative 1, so let's see. The first point we want to
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plot is (0,1). The next point, quarter point, is (pi over 6, 0), (pi over 3, -1), (pi over 2, 0), and (2 pi over 3, positive 1).
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So here is one period for y equals cosine 3x. What is its amplitude? Well we can see that it's still 1. Notice that,
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in this case, A is 1, so the amplitude is 1. What is the range? Well it's negative 1 up to positive 1.
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Let's look at graphing functions of the form y equals A sine Bx and y equals A cosine Bx. The amplitude of these
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functions will be the absolute value of A. The period will be 2 pi over B, again for B positive. Let's look at this
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example y equals sine negative pi x. Right off the bat we notice that our B is negative. So what do we do? Well, we
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have to remember something about the sine function, and that is that the sine is odd. And that means that the
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sine of negative x is the opposite of the sine of x, so we can rewrite this function as negative sine of pi x.
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Now we can find our amplitude which will be the absolute value of negative 1, which is 1, and our period
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for this function will be 2 pi divided by pi, which is 2. Now we want to get the quarter points for this function. We need
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to get one fourth of the period to figure out the interval between the quarter points, which will be 1/2. So we're going
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to be adding 1/2 to each of our x-coordinates to get the x-coordinates for y equals negative sine pi x. The first
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x-coordinate will stay 0. Let's just do the x-coordinates first and then we'll come back and do the y-coordinates.
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So what we want to do is we want to add 1/2 to each of these x-coordinates. So 0 plus 1/2 is 1/2. 1/2 plus 1/2 is 1.
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1 plus 1/2 is 3/2. And then 3/2 plus another 1/2 gives you 2. So that's our x-coordinates for our quarter points.
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Now how do we get the y-coordinates? Remember the y-coordinates are affected by A, so we want to be multiplying
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these y-coordinates by A, which in this case is negative 1. So we're going to multiply each of these y-coordinates
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by negative 1. So let's see. We're going to get 0. 1 times negative 1 is negative 1. 0 times negative 1 is 0.
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Negative 1 times negative 1 is positive 1, and then 0 times negative 1 is 0. So here are our quarter points
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for y equals negative sine pi x. Let's graph that function.
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Ok, so let's see. We have (0, 0), and then we have ... oops, not pi over 2. It should have been 1/2. So all of these
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should be 1/2, 1, 3/2, and 2. So this point (1/2, -1) down here. (1/2, 0) I mean sorry, (1, 0). (3/2, 1) and then (2, 0).
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So here would be one period for the function y equals negative sine pi x. Notice that it's going down first.
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There's the negative which reflects it about the x-axis. What's the amplitude? We said that was 1.
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What's the range for this function? Let's write that down. The range would be negative 1 to positive 1.
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Let's take a look at graphing y equals negative 2 cosine of negative 4x. Again, the first thing I need to notice is
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when B is negative. To find the period you need a positive B. So we need to rewrite this function. Remember that
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the cosine is even. What does that mean? It means that the cosine of negative x is the same as the cosine of x,
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which means the cosine of negative 4x is the same as the cosine of 4x. So this can be rewritten as y equals
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negative 2 cosine of 4x. Now, what's the amplitude? Well the amplitude is going to be the absolute value
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of negative 2 which of course is positive 2. What's the period? Well we find the period by taking 2 pi dividing by B,
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which in this case is 4, so our period will be pi over 2. Now how do we get our x-coordinates? We need to know
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one fourth of the period. And what is one fourth of pi over 2? Well, that's pi over 8. So pi over 8 is going to be what
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we add to each x-coordinate to get our x-coordinates for our quarter points for this function.
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So let's write those down. So let's see, we have negative 2 cosine 4x. The first x-coordinate will stay 0.
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Now what do we add to 0 to get to the next one? You add pi eighths. So this next x-coordinate will be pi over 8.
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Again, you're adding pi over 8 to each x-coordinate. What's this x-coordinate going to be? Well, what is pi over 8
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plus pi over 8? Well, that's 2 pi over 8 which you reduce to pi over 4. And then keep adding pi over 8.
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That's 2 pi over 8 plus pi over 8 gives you 3 pi over 8. And then 3 pi over 8 plus another pi over 8 gives you
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4 pi over 8, which of course is pi over 2. So those are our x-coordinates for our quarter points. How do we get our
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y-coordinates? Well, we need to multiply each of these y-coordinates by A, which is negative 2. Make sure you
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don't multiply by 2. It's always A, not the amplitude. So let's see. What is 1 times negative 2? Negative 2.
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0 times negative 2 is 0. Negative 1 times negative 2 is a positive 2. 0 times negative 2, 0. And 1 times negative 2,
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negative 2. So here are our quarter points for y equals negative 2 cosine 4x. Let's look at that graph.
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So we have (0, -2), (pi over 8, 0), (pi over 4, 2), (3 pi over 8, 0), and (pi over 2, -2). So here would be one period for
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this function y equals negative 2 cosine 4x. We figured out the amplitude for this function was 2.
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What's the range for this function? Well the range is going to be negative 2 to 2. Remember it's always
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negative absolute value of A to positive absolute value of A.
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In this example, we're given a graph and we want to determine the equation of a function of the form
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y equals A sine Bx or y equals A cosine Bx. So we have to decide whether we're going to use a sine function or a
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cosine function. Well, how do we know? Let's look at the first quarter point. It's (0, 0). It goes through the origin, so
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we know we're going to use a sine function. So we're going to write y equals A sine Bx, so now we have to find
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B and A. Well remember B is with the period, so let's look at this graph. What's the period for this graph?
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We can see the period is 8 pi, so let's write that down. And how do we get the period? Do you remember?
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That's right. You take 2 pi and you divide by B to get the period. So we can solve for B. Multiply both sides by B.
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So we get 2 pi divided by 8 pi, which is 1/4. So now we know B is 1/4. Now we need to find A. Well I can see from
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the graph that the amplitude is 2, but that doesn't mean that A is 2. Notice that this graph is decreasing first.
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This y-coordinate of this quarter point is negative 2, so I know A is negative 2. So now what's our sine function?
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y equals negative 2 sine of 1/4 times x.
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You need to practice working these examples until you've mastered the skills and concepts involved.