WEBVTT mathematics/trigonometry/murray
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Hi this is educator.com and today we're going to learn about modified sine waves.
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We're going to learn how to analyze and graph these functions given by, instead of just talking about sin(x) and cos(x), now we're going throw a whole bunch of constants in there.
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We'll talk about asin(bx+c) and then throw a constant outside +d and the same kind of thing with cosine.
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That takes a basic sine or cosine graph and it moves it all around.
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We're going to learn some vocabulary to describe those movements and we're going to learn how to graph those.
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First of all, some vocabulary, remember we're talking about the graph asin(bx+c)+d.
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We're talking about sine waves, these are functions that basically have this shape like sin(x), but they maybe moved in different waves.
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We need some vocabulary to describe the different ways that could be moved.
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The first one that we're going to learn is amplitude.
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The amplitude of the sine waves is the vertical distance from the middle of the waves to the peaks.
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What that represents graphically is this distance right here, that's the amplitude.
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Of course, that's the same as this distance, that's also the amplitude but you can measure it either way.
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If the wave is moved up, if it's floating up above, the x axis, somewhere like that, then the amplitude is still the distance from the middle of the wave to the peaks.
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In terms of equations, it's very easy to spot the amplitude.
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When you're given asin(bx+c)+d, the amplitude is just that number a, or if the a is negative, you just take the absolute value.
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It's just the positive version of that number a that tells you the amplitude right there.
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It tells you how far up away it's going to the peaks, how far down it's going to the valleys.
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The period of a sine waves, I'll show this in red, is the horizontal distance for the wave to do one complete cycle from one peak to the next peak.
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That is the period right there, of that wave, that's the period.
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On that one, that's the period right there.
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Remember, when you're working out the equations, remember that if you have sin(x) that has period 2π, it takes 2π to repeat itself.
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We learned that when we looked into the original sine graphs.
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If you have sin(2x), that makes it wobble up and down twice as fast, the period would be πsin(4x), the period would be π/2.
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The pattern that you noticed here is it that the period is given by 2π over the coefficient of x, 2π/b.
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The b there tells you where the period is, not the b itself but you plug b into that equation and that tells what the period is.
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Two more vocabulary words we need to learn, the phase-shift of a sine wave is the horizontal distance that the wave is shifted from the traditional starting position.
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Let me rewrite the equation here, asin(bx+c)+d.
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The traditional starting position for sine would be at (0,0), and the traditional starting position for cosine would be at (0,1).
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Those are the traditional starting positions but the phase-shift will move the graph to the right or the left.
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In these equations, it's given by -c/b, and that seems a little mysterious and let me explain that a little bit.
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We can write this bx+c, first of all, we can factor b out, we can write that b[x + (c/b)].
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Then, we can write that as b[x - (-c/b)].
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That's where that -c/b comes from.
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If you have trouble remembering that formula -c/b, you can go through this little process to remember that, x-(c/b), that shows you that it moves it c/b units to the right.
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Let me draw that, I'll draw that in blue.
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If you're starting with a sine curve, the phase-shift is the amount that it moves over, that's the phase-shift right there, it moves it over -c/b units.
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If you're starting with a cosine curve, that's that phase-shift right there.
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Finally, the vertical shift is what happens when you take the graph and you just move it up or down vertically without changing anything else.
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Let me start with a sine curve.
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If we apply a vertical shift to that, that amount right there, I'll draw this in red, that's the vertical shift.
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That's a little bit easier to pick out than some of the others, because that's just the d in the original equation.
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If you're moving the graph up or down by an amount of d.
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This can get pretty tricky we're starting with the basic sine and cosine curves, but then we're moving around and stretching them out, and moving them up and down in all different ways.
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It's a little bit tricky but we'll go through some of the examples and you'll get the hang of it.
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In our first example here, we're given an equation 3cos(4x+π)+2.
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We have to identify all these various things, the amplitude, the period, the phase-shift and the vertical shift and then we're going to draw a graph of the function.
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The key thing here is it if you identify these things in order, then it becomes very easy to pick them out using the equations.
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The graph isn't too hard as long as you these things in order.
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The amplitude, remember, that's just the number on the outside, the a in the original equation.
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Let me write down the original equation, acos(bx+c)+d.
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The amplitude is just the a right there, that's the 3.
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The period is 2π/b, the b there is 4, that's 2π/4 which is π/2.
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The phase-shift is -c/b, our c here is π, -π, b is 4, that's -π/4.
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The vertical shift is just that last term d, which is 2.
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Those are the answers to the first part of the question.
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Trickier part is doing the graph and there's a general strategy for doing this graphs that sort of always works, but you really have to follow it closely.
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The strategy is to start with the basic cosine graph, which hopefully you remember how to do it, you start with the basic cosine graph.
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Then move it around according to each one of these parameters.
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The key thing here is you have to do it in order, you have to do amplitude period phase-shift then vertical shift.
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Let's see how that works out.
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Let me draw a basic cosine graph and then we'll try moving it around according to these different parameters.
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Remember, basic cosine graph, there's π, there's 2π, π/2, 3π/2.
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Basic cosine graph starts at 1, goes down to 0 at π/2, -1 at π, up to 0 at 3π/2, and it's 1 again at 2π.
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That's the basic cosine graph, you pretty much have to remember that to get started here.
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First thing we're going to do, is we're going to change the amplitude.
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Let me keep track of this as I go along.
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First one we graphed was cos(x), just y=cos(x).
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Next we're going to graph is 3cos(x), we're going to bring in the amplitude.
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What that does is that it stretches up the peak, then it stretches down the valleys by a factor of 3.
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I'm going to draw the same shift graph but three times as tall, and three times as deep.
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Instead of going from 1 to -1, goes down to -3 and up to 3.
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That second graph I drew there was 3cos(x).
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The next one is to introduce is the period.
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The period is supposed to be π/2, remember, the period is the amount of horizontal distance between one peak and the next peak.
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My current peak is...
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I'm going to change the period to π/2.
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What I'm really graphing here is 3cos4x.
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What that's going to do is, instead of having a period 2π, it shrinks it horizontally, or it compresses it horizontally so that it does a complete period in the space of π/2.
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There's π/2 right there.
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I need to do a whole period between there and there.
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Every π/2, it does a complete cycle.
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What I just drew was 3cos4x.
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The phase-shift I'm going to introduce is -π/4, that means it moves it π/4 to the left and it's getting a little bit messy.
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I'm going to see if I can draw this in red or will see if it's still visible.
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Instead of going from 0 to π/2, I'm going to draw my graph from -π/4 to π/4, because we're moving it to the left by π/4.
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That's the graph in red there, 3cos4x+π because I've introduced the phase-shift in there.
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Now, it's really going to get messy if I try to draw any more on the same axis, so I'm going to set up a new set of axis.
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We had π/2, π, -π/2.
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I'll redraw the red one on this set of axis, -π/2, and that one is going from 3 to -3.
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Remember the red one is the previous graph shifted over by -π/4.
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Finally, we need to introduce the vertical shift of 2, and I'll do this last graph in blue.
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That takes the entire graph and raises it up by 2 units.
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That means instead of going to 3 from -3, it's going to go up in π, and instead of going down to -3, it only goes down to -1.
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Let me label that more clearly, -1.
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I'll draw this one in blue, this is now 3cos4x+π, and I'm introducing the vertical shift of +2.
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I'm taking the graph and I'm moving it up 2 units there.
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That blue graph at the end is our final function.
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This is really a pretty complicated process.
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There's a lot of steps involved but each individual step is not that hard, and if you do them in order and you're careful about each one, it's not too bad.
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Let me just recap there, we started with the original graph of cos(x), that's the starting point.
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Then we introduced the amplitude, and that stretches it vertically by a factor of 3, stretches it up and down.
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We introduced the period which compresses it horizontally.
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We introduced the phase-shift, which takes the whole thing and it moves it to the right or the left.
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Finally, We introduced the vertical shift which takes the whole thing and moves it up or down.
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Remember, it's important to do these in order, amplitude, period, phase-shift, vertical shift.
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If you do those out of order, then they'll mess each other up as you go along.
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We really want to do those in order.
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We want to practice several of these, so let's get moving in other example.
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Here's another example, same questions here, amplitude, period, phase-shift, and vertical shift of the following function.
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Remember we can read this off quickly, just remembering the formula asin(bx+c)+d.
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If we can figure out what a, b, c, and d are, we have formulas for all of these properties.
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Amplitude, that's the a, or if the a is negative, make it positive, that's the absolute value of a, which is just 2 here.
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The period is 2π/b, the b is 2 here, so that's π.
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The phase-shift is -c/b, which is, okay, c is -π/3, so negative of that is (π/3)/2 will give us π/6.
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Finally, the vertical shift is d, which ,in this case, is just 0.
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Finally, the fun part, we get to graph the function
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Remember, you always start with your basic sine or cosine graph and then you start moving it around according to these different parameters but you got to keep these parameters in order.
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Let me start with this one's a sine graph.
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I know the basic shape of the sine graph, I've got that memorized, π and 2π.
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I know sine always starts at 0, goes up to 1, comes back down to 0, to -1, and then back to 0.
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There's 1, -1, there's π, π/2, 3π/2, so that's my basic sine graph.
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That's the first thing I graphed there, sin(x), remember this is the graph of sine and cosine x.
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Now, we start introducing these other attributes and it's important to go in order.
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First of all, we're going to introduce the amplitude.
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The amplitude is 2, but we're really multiplying the graph by -2, -2sin(x) is what I'm going to graph next.
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-2sin(x) that stretches it vertically because of the 2, but it also flips it vertically.
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Instead of starting by going up, it's going to go down, goes down to -2, up to 0, up to 2, and back down to 0.
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That's first one was a little lop-sided, let me just see if I can make that a little more, a little smoother.
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Okay, we've got -2sin(x), look it's got a bigger amplitude than the original graph and it's flipped over because of the negative sign.
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Next thing I'll introduce is the period.
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The period is supposed to be π, so I'm graphing -2sin(2x).
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That speeds the whole thing up, it shortens the period because the period is now π instead of 2π.
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I need to do that entire graph in the space of π instead of 2π.
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There, that one that I just graphed, I shortened the period to be π instead of 2π.
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The period is now π on that new graph.
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Next, we're going to do the phase-shift, that's π/6 units to the right.
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The phase-shift, what I'm about to graph is -2sin(2x)-π/3, so that takes the whole graph and it shifted over π/3 units to the right.
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Let me do this one in red.
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I'm going to take that last graph and shift it over π/3 units to the right.
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Instead of starting at (0,0), it starts at π/3, it's going to come back down to 0, at 5π/6.
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The phase-shift is supposed to be π/6, so I'm going to move everything over by π/6.
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Instead of starting at 0, I'm going to start at π/6 and come back at (π/2)+(π/6) which is actually 2π/3.
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That red curve that I graphed there is -2sin(2x)-π/3.
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The last step is to do the vertical shift which is 0, so we don't have to move the graph at all which means we're done.
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This last graph is the one we want.
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Again, it's a matter of breaking these equations down into their parts.
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It's very complicated if you kind of look at the whole thing but if you look at each steps and you keep each steps in order then it's not to hard.
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Remember the equations for amplitude, period, phase-shift, and vertical shift.
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Once you've got those, you start with your basic sin(x) graph then you change the amplitude which stretches it out up and down, or might flip it.
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You change the period which compreseses it horizontally.
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You do the phase-shift which takes the whole thing and without compressing it, it moves it to the right or the left.
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Finally, the vertical shift moves it up or down.
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You could just keep moving these graphs around until you build up the equation you're looking for.
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This one's a little bit different, we're asked to find the sine wave.
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This time we're told what all the properties are.
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We're given the amplitude, the period, the phase-shift, and the vertical shift.
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We want to find an equation and we want to graph the function.
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I'm going to kind of build this up, the same way we're building the earlier graphs.
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I'm going to start with the basic sin(x), so that's my basic sine wave.
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Now, I'm going to give an amplitude 2, and remember 2 is just a number on the outside, the a.
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Let me rewrite that equation, asin(bx+c)+d.
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Amplitude 2 means a is 2, 2sin(x).
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Now, period 4π, remember our equation for period was 2π/b, that is supposed to be equal to 4π.
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When you solve that out, that tells us that b=1/2, so the b has to be equal to 1/2.
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That means our equation is now 2sin(1/2)x, so we've incorporated the period.
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Phase-shift is supposed to be π/2, but remember the phase-shift, our formula for that is -c/b.
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b is already, we figured out as 1/2, so that's -c/(1/2).
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If we do a little bit of algebra here, we get (1/2)π=2c, so c=-π/4.
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I got that from the equation -c/b=π/2.
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I already figured out my b, now I can figure out my c.
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The next part of that equation is 2sin[(-1/2)x-(π/4)].
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Finally, I want to talk about the vertical shift, which is supposed to be 1 and that's the d.
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Finally, my equation is 2sin[(1/2)x-(π/4)]+1.
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That is the sine wave that I'm looking for.
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Now, I want to graph that thing.
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I start out with the basic sine wave, π, 2π, π/2, 3π/2.
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Basic sine wave starts at 0, goes up to 1, back down to 0 at π, down to -1, back up to 0 at 2π, basic sine wave.
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Now, I'll introduce these properties in order.
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I'll start out with the amplitude.
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Amplitude's supposed to be 2, so I'm going to stretch this thing up instead of going from 1 to -1, it's going to go up from 2 up to 2, and down to -2.
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It'll stretch the thing up, and down.
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Period is 4π, that means the thing stretches out, so that it'll only does one cycle every 4π.
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That means I have to extend my graph quite a bit here, 3π, 4π, so I'll stretch the thing out, so it'll only does the cycle every 4π.
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Now, phase-shift π/2, that means the thing is going to shift π/2 units to the right.
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I better draw a new set of axis here.
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I've got π, 2π, 3π and 4π, 5π.
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What I want to do is take the graph I have above and move it π/2 units to the right because we have phase-shift π/2.
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Instead of starting at 0, I'm going to start at π/2, I go up to there, back down to 0 at 2π plus π/2, back down to -2 between 3 and 4π and back up to 0 there.
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So, connect these up.
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Finally, I have to do one with vertical shift 1, that means I'll take the whole graph and I'll move it up by one unit.
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That means instead of going or peak at 2, it's going to peak at 3 now.
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Instead of going down to -2, it gets moved up by 1 unit so it's going to go down to -1.
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Let me draw this last final curve in red.
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Everything gets moved up by 1 unit.
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I'm going to plot some points here, moving everything up by 1 unit.
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That final curve is the one we want.
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That's 2sin[(1/2)x-(π/4)]+1.
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Again, it's a complicated procedure but if you take it step by step, each one of the steps is not too hard.
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First, we kind of reconstructed the equation from these parameters that we were given.
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Basically we figured out a, b, c and d from these parameters that we were given by sort of reverse engineering the formulas 2π/b, -c/b, and the d.
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Then, we went through step by step.
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We started with the basic sine curve.
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We changed its amplitude, stretched it vertically.
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We changed its period which stretches out horizontally.
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We changed its phase shift which moved it over horizontally not stretching but just moving it without stretching.
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Then, we did the vertical shift, moving it up or down.
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You should practice a few of these curves on your own.
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We'll come back and try some more examples together later.