For more information, please see full course syllabus of Trigonometry
For more information, please see full course syllabus of Trigonometry
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D
Main definitions and formulas:
 The amplitude of a sine wave is the vertical distance from the middle of the waves to the peaks (or from the middle to the valleys). In the equations above, it is given by  A .
 The period of a sine wave is the horizontal distance for the wave to do one complete cycle from one peak to the next peak. In the equations above, it is given by (2π/B).
 The phase shift of a sine wave is the horizontal distance the wave is shifted from the traditional starting position. In the equations above, it is given by − (C/B).
 The vertical shift of a sine wave is the vertical distance that the middle of the wave is shifted from the xaxis. In the equations above, it is given by D.
Example 1:
Identify the amplitude, period, phase shift, and vertical shift of the following function, and graph the function:

Example 2:
Identify the amplitude, period, phase shift, and vertical shift of the following function, and graph the function:

Example 3:
Find a sine wave with amplitude 2, period 4π , phase shift (π/2), and vertical shift 1. Graph the function.Example 4:
Identify the amplitude, period, phase shift, and vertical shift of the following function, and graph the function:

Example 5:
Find a cosine wave with amplitude 2, period 3π , phase shift (π/2), and vertical shift 2. Graph the function.Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D
[1/2]cos(3x + [(π)/6]) − 3
 Recall: Acos(Bx + C) + D where A is the amplitude; [(2π)/B] is the period; − [C/B] is the phase shift; D is the vertical shift
 A = [1/2], B = 3, C = [(π)/6], D =  3
− 4sin(2x − [(π)/4]) + 1
 Recall: Asin(Bx + C) + D where A is the amplitude; [(2π)/B] is the period; − [C/B] is the phase shift; D is the vertical shift
 A =  4, B = 2, C = − [(π)/4], D = 1
 Recall: Acos(Bx + C) + D where A is the amplitude; [(2π)/B] is the period; − [C/B] is the phase shift; D is the vertical shift
 A =  2, D = 4, We need to find B and C by working backwards
 First find B by using the period. We know Period = [(2π)/B] so,
 [(π)/3] = [(2π)/B] ⇒ Bπ = 6π ⇒ B = 6
 Now find C by using the phase shift and the value of B we just calculated
 π = − [C/6] ⇒ 6π =  C ⇒ C =  6π
 Recall: Asin(Bx + C) + D where A is the amplitude; [(2π)/B] is the period; − [C/B] is the phase shift; D is the vertical shift
 A = [1/2], D =  2, We need to find B and C by working backwards
 First find B by using the period. We know Period = [(2π)/B] so,
 4π = [(2π)/B] ⇒ B4π = 2π ⇒ B = [1/2]
 Now find C by using the phase shift and the value of B we just calculated
 [(π)/4] = − [C/([1/2])] ⇒ [1/2]π = − 4C ⇒ C = − [(π)/8]
[1/3]sin([1/2]x − [(π)/3])  5
 Recall: Asin(Bx + C) + D where A is the amplitude; [(2π)/B] is the period; − [C/B] is the phase shift; D is the vertical shift
 A = [1/3], B = [1/2], C = − [(π)/3], D =  5
3cos(x + π)  3
 Recall: Acos(Bx + C) + D where A is the amplitude; [(2π)/B] is the period; − [C/B] is the phase shift; D is the vertical shift
 Amplitude = A = 3, Period = [(2π)/B] = [(2π)/1] = 2π, Phase Shift = − [C/B] = − [(π)/1] = − π, Vertival Shift = D =  3
 First graph f(x) = cos(x)
 Now graph the amplitude f(x) = 3cos(x)
 Now graph the period. Since the period is 2π, the graph will stay the same
 Now graph the phase shift f(x) = 3cos(x + π). The graph will shift p units to the left
 Now graph the vertical shift f(x) = 3cos(x + π)  3. The graph will shift down from 3 to 0 and  3 to  6
2sin(2x  [(π)/3]) + 2
 Recall: Asin(Bx + C) + D where A is the amplitude; [(2π)/B] is the period; − [C/B] is the phase shift; D is the vertical shift
 Amplitude = A = 2, Period = [(2π)/2] = [(2π)/2] = π, Phase Shift = − [C/B] = − [( − [(π)/3])/2] = [(π)/6], Vertival Shift = D = 2
 First graph f(x) = sin(x)
 Now graph the amplitude f(x) = 2sin(x)
 Now graph the period. The period is p, so the graph will cycle every p for f(x) = 2sin(2x)
 Now graph the phase shift f(x) = 2sin(2x  [(π)/3]). The graph will shift [(π)/6] units to the right
 Now graph the vertical shift f(x) = 2sin(2x  [(π)/3]) + 2. The graph will shift up from 2 to 4 and  2 to 0
 cos([1/2]x − [(π)/4]) + 2
 Recall: Acos(Bx + C) + D where A is the amplitude; [(2π)/B] is the period; − [C/B] is the phase shift; D is the vertical shift
 Amplitude = A = 1, Period = [(2π)/2] = [(2π)/([1/2])] = 4π, Phase Shift = − [C/B] = − [( − [(π)/4])/([1/2])] = [(π)/2], Vertival Shift = D = 2
 First graph f(x) = cos(x)
 Now graph the amplitude f(x) =  cos(x)
 Now graph the period. The period is 4π, so the graph will cycle every 4π for f(x) =  cos([1/2]x)
 Now graph the phase shift f(x) =  cos([1/2]x  [(π)/4]). The graph will shift [(π)/2] units to the right
 Now graph the vertical shift f(x) =  cos([1/2]x  [(π)/4]) + 2. The graph will shift up from  1 to 1 and 1 to 3
 3sin(4x + 4π)  1
 Recall: Asin(Bx + C) + D where A is the amplitude; [(2π)/B] is the period; − [C/B] is the phase shift; D is the vertical shift
 Amplitude = A = 3 ,Period = [(2π)/B] = [(2π)/4] = [(π)/2], Phase Shift = − [C/B] = − [(4π)/4] = − π, Vertival Shift = D =  1
 First graph f(x) = sin(x)
 Now graph the amplitude f(x) =  3sin(x)
 Now graph the period. The period is [(π)/2], so the graph will cycle every [(π)/2] for f(x) =  3sin(4x)
 Now graph the phase shift f(x) =  3sin(4x + 4π). The graph will shift π units to the left
 Now graph the vertical shift f(x) =  3sin(4x + 4π)  1. The graph will shift down from 3 to 2 and  3 to  4
 Recall: Acos(Bx + C) + D where A is the amplitude; [(2π)/B] is the period; − [C/B] is the phase shift; D is the vertical shift
 A = 4, D = 7, We need to find B and C by working backwards
 First find B by using the period. We know Period = [(2π)/B] so,
 [(π)/4] = [(2π)/B] ⇒ Bπ = 8π ⇒ B = 8
 Now find C by using the phase shift and the value of B we just calculated
 [(π)/3] = − [C/8] ⇒ 8π =  3C ⇒ C = − [(8π)/3]
*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.
Answer
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D
Lecture Slides are screencaptured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
 Intro
 Amplitude and Period of a Sine Wave
 Phase Shift and Vertical Shift
 Example 1: Amplitude/Period/Phase and Vertical Shift
 Example 2: Amplitude/Period/Phase and Vertical Shift
 Example 3: Find Sine Wave Given Attributes
 Extra Example 1: Amplitude/Period/Phase and Vertical Shift
 Extra Example 2: Find Cosine Wave Given Attributes
 Intro 0:00
 Amplitude and Period of a Sine Wave 0:38
 Sine Wave Graph
 Amplitude: Distance from Middle to Peak
 Peak: Distance from Peak to Peak
 Phase Shift and Vertical Shift 4:13
 Phase Shift: Distance Shifted Horizontally
 Vertical Shift: Distance Shifted Vertically
 Example 1: Amplitude/Period/Phase and Vertical Shift 8:04
 Example 2: Amplitude/Period/Phase and Vertical Shift 17:39
 Example 3: Find Sine Wave Given Attributes 25:23
 Extra Example 1: Amplitude/Period/Phase and Vertical Shift
 Extra Example 2: Find Cosine Wave Given Attributes
Trigonometry Online Course
I. Trigonometric Functions  

Angles  39:05  
Sine and Cosine Functions  43:16  
Sine and Cosine Values of Special Angles  33:05  
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D  52:03  
Tangent and Cotangent Functions  36:04  
Secant and Cosecant Functions  27:18  
Inverse Trigonometric Functions  32:58  
Computations of Inverse Trigonometric Functions  31:08  
II. Trigonometric Identities  
Pythagorean Identity  19:11  
Identity Tan(squared)x+1=Sec(squared)x  23:16  
Addition and Subtraction Formulas  52:52  
Double Angle Formulas  29:05  
HalfAngle Formulas  43:55  
III. Applications of Trigonometry  
Trigonometry in Right Angles  25:43  
Law of Sines  56:40  
Law of Cosines  49:05  
Finding the Area of a Triangle  27:37  
Word Problems and Applications of Trigonometry  34:25  
Vectors  46:42  
IV. Complex Numbers and Polar Coordinates  
Polar Coordinates  1:07:35  
Complex Numbers  35:59  
Polar Form of Complex Numbers  40:43  
DeMoivre's Theorem  57:37 
Transcription: Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D
Hi this is www.educator.com and we are going to try more examples of modified sin waves where we start with the basic equation of sin(x) or cos(x) and the graph of sin(x) or cos(x).0000
We introduce these constant which are going to change some of its attributes and we see what that does to the graph.0013
Remember the equation we are working with in general is (a)sin arcos(bx + c) + d and then from each of those values we figured out these various attributes amplitude, period, phase shift and vertical shift.0021
In this particular equation, the amplitude, remember that is just (a) so you read that as 4, the period is 2pi/b , the b is 2 here, so that is 2pi/2 which is pi.0041
The phase shift, that is the strangest one â€“c/b, that is (â€“pi/2)/2 or â€“pi/4.0066
Finally the vertical shift, that is the easier one is 1 here.0085
Again we will start out with the basic sin way, we will work through introducing these attributes one at a time and see how that moves around and create a new function for us.0097
Let us start out with the basic sin way.0108
There is pi, 2pi, so Iâ€™m just going to graph sin(x) to start with.0119
Iâ€™m going to graph it flat because Iâ€™m looking ahead and noticing that the next step is to increase the amplitude to 4.0130
It is really going to be stretched up, Iâ€™m going to keep my scale narrow here.0139
The next step is to introduce the amplitude, we are going to graph 4sin(x).0148
That stretches it up and down by a factor of 4, 4 goes down to 4 goes up to +4, that makes it a lot steeper.0156
That is 4sin(x), we got the amplitude incorporated there, remember it is very important to do these in order, amplitude, period, phase shift, vertical shift.0175
Next step is to introduce the period, right now the period is 2pi and we want the period to be pi.0185
Instead of doing a full cycle in 2 pi units, that is going to do cycle in pi units, that means it shuffles twice as fast.0197
Let me try and draw that.0205
That is pi/2 there, 3pi/2, ok we got the period incorporated.0220
What I really graphed there was 4sin(2x), next we want to incorporate the phase shift.0230
I will be graphing 4sin(2x) + pi/2 and this is starting to get a little crowded here, so Iâ€™m going to do this one in red.0241
That means that the phase shift is â€“pi/4, that means we move â€“pi/4 units to the left.0257
Instead of starting at 0, Iâ€™m going to start at â€“pi/4 and it came back down before pi/2 so now it is going to come back at pi/4.0267
It used to come back up at pi and now it comes back up at 3pi/4.0287
That red graph is what we get by incorporating the phase shift and finally we will incorporate the vertical shift and I will do this one on blue.0301
We are going to be doing 4sin(2x + pi/2) 1, that means we take the graph and we move it down by one unit.0316
It is not the same as changing the amplitude where things stretch up and down.0328
Here we are not stretching anything, we are just moving everything directly down by one unit.0331
Instead of starting at 0 going up to 4 down to 4, everything is moved down by 1 unit.0336
It will peak at three and instead of going down to 4, it will go down to 5 and the middle part will be at 1.0346
Let me go and try to graph that in blue.0355
It comes back to 1 instead of 0, bottoms out at 5 instead of 4, goes back up peaks at three, back down to 5 and back up to 1 again.0362
This blue curve is the final graph there, it is very complicated by the time it is all over but it is a bunch of simple steps.0382
The first step is to look at the equation and identify these quantities, amplitude, period, phase shift, and vertical shift.0395
We got pretty simple equations for each one of those, the tricky part of the graphing is where you start with sin(x).0403
You remember how to graph sin(x), that is very easy and then start incorporating these attributes one by one.0414
Amplitude stretches it vertically up and down, period stretches or compresses it horizontally, in this case the period was pi and since it was 2pi/4 we have to compress it by a factor of 2.0420
Phase shift moves it over to left and right, vertical shift moves it up or down.0434
We finally end up with this blue curve that has been modified according to all those attributes.0440
Now we are being asked to find a cos wave and we are given the attributes but we are not given the equations.0000
We have to reverse engineer the equation from this attributes.0007
Remember the equation we are going to go for is (a)cos(bx+c)+d, but we do not know what a, b, c, and d are, we got to figure them out from these attributes.0012
Amplitude is the opposite value of a, we will take a=2 to give ourselves amplitude =2.0028
The period is 2pi/b and that is supposed to be equal to 3pi.0048
From that we can figure out if we solve that for b, we get (b) is equal to 2/3, that is how we can figure out what (b) should be.0063
The phase shift is â€“c/b which is supposed to be equal to pi/2, now that requires a little bit of solving, so we get â€“c/b we already figure out that was 2/3 is equal to pi/2.0073
That is 3(c)/2 is equal to pi/2, the 2 is cancel and so we get (c) is equal to â€“pi/3.0106
That is how we figured out what (c) is, we reversed engineer this equation â€“c/b.0122
Finally the vertical shift, is (d) which is 2, we figured out what a, b, c, and d are, that means that our equation is a=2cos(2/3x)(pi/32).0128
We figured out the first part of the problem which is to find the equation, now we are going to do the tricky part which is graphing it.0162
It is not so tricky if you start with the basic cos wave if you remember how to graph that and then you introduce this attributes in the right order.0168
I will start with the basic cos wave, cos(x), there is pi, 2pi, 3pi. Cos(x) I know it starts at 1, there is 1, 1, it starts at 1, it goes down to 0.0179
Bottoms out at 1 goes back up to 0, goes back up to 1, now I got cos(x) that I pretty much do from memory.0217
Next Iâ€™m going to do is introduce the amplitude, Iâ€™m going to talk about 2cos(x) that just stretches it vertically up and down, it goes up to 2 and down to 2.0228
Next Iâ€™m going to adjust the period, what Iâ€™m actually going to be graphing is 2cos(2/3x).0257
I know that makes the period 3pi, our original period is 2pi, this is stretching it out by a factor of 50% instead of doing a full cycle in 2pi, it is going to do a full cycle in 3pi.0269
Let me give my self some coordinates here, it still going to start at 0 but it is going to bottom out now.0286
It is going to finish up at 3pi instead at 2pi, it is going to bottom out halfway between them that is 3pi/2.0295
I took the graph and I stretched it out so that it has a period of 3pi now, phase shift â€“pi/3.0324
What Iâ€™m going to be graphing is 2cos(2/3x)pi/3, sorry the phase shift is pi/2, the c value is â€“pi/3 but the phase shift is pi/2.0338
Iâ€™m going to take this graph and move it over pi/2 units to the right.0358
It looks like it is getting a little complicated now, Iâ€™m going to switch into red here.0364
We are going to graph the phase shift in red so that means instead of starting at 0, Iâ€™m going to start at pi/2.0371
It is going to bottom out at 2pi and it is going to come back up at 3pi + pi/2.0379
My orientation points were pi/2, 2pi, 3pi + pi/2, because I just took my orientation points before and move them all over by pi/2.0413
That incorporated the phase shift of pi/2, we are almost done here.0425
We got on more step to incorporate which is this vertical shift of 2.0430
Iâ€™m going to do this one in blue so we are going to be graphing 2cos(2/3x)(pi/32) that means we are going to take the whole graph and just move it down 2 units.0435
Now remember, before our peaks and values were 2 and 2, we moved that down by 2 units and the peaks and values are now going to be at 0 and 4.0455
Iâ€™m going to extend my axis down to 4, 3, 4.0467
Iâ€™m going to take my reference points from before and move them down by 2 units.0476
This graph peaks at 0 and it never actually goes north of the x axis because it is going up to 0 and down to 4.0505
That is the final graph that we have been asked for, let us recap there.0520
We are given all these attributes amplitude, period, phase shift, and vertical shift.0528
We know the equation for those 4 things, absolute value of a, 2pi/b, c/b and d.0533
What we do is we plug in those attributes and reverse engineer to figure out what a, b, c, and d are.0541
Then we can put those together to get the equation of the function that we are trying to graph.0550
To actually graph it, we start with basic cos curve, we introduced the amplitude.0557
It is very important to do this in order, introduce the amplitude where you expanded it up, up and down.0563
The thing actually stretches out, we introduced the period which collapses it horizontally or stretches it out horizontally.0570
In this case, we are changing the period from 2pi to 3pi, so that stretches it horizontally.0581
We introduced the phase shift which moves it horizontally to the right or left without stretching it.0587
The period was the part where you stretch it, the phase shift just moves it without stretching it.0593
Finally, we introduced the vertical shift which moves it up and down without stretching it, in this case it moved down.0599
At the end of it, it is quite a complicated process but individually each one of these steps if you keep them in order is not to hard.0608
You just start with you original cos graph and then move them around according to these steps until you get the one you want.0615
That is the end of our lecture on modified sin waves, this is www.educator.com.0622
Hi this is educator.com and today we're going to learn about modified sine waves.0000
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.0007
We'll talk about asin(bx+c) and then throw a constant outside +d and the same kind of thing with cosine.0020
That takes a basic sine or cosine graph and it moves it all around.0027
We're going to learn some vocabulary to describe those movements and we're going to learn how to graph those.0030
First of all, some vocabulary, remember we're talking about the graph asin(bx+c)+d.0038
We're talking about sine waves, these are functions that basically have this shape like sin(x), but they maybe moved in different waves.0057
We need some vocabulary to describe the different ways that could be moved.0073
The first one that we're going to learn is amplitude.0079
The amplitude of the sine waves is the vertical distance from the middle of the waves to the peaks.0082
What that represents graphically is this distance right here, that's the amplitude.0089
Of course, that's the same as this distance, that's also the amplitude but you can measure it either way.0102
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.0114
In terms of equations, it's very easy to spot the amplitude.0129
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.0133
It's just the positive version of that number a that tells you the amplitude right there.0148
It tells you how far up away it's going to the peaks, how far down it's going to the valleys.0158
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.0160
That is the period right there, of that wave, that's the period.0179
On that one, that's the period right there.0189
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.0198
We learned that when we looked into the original sine graphs.0212
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/2.0215
The pattern that you noticed here is it that the period is given by 2π over the coefficient of x, 2π/b.0230
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.0242
Two more vocabulary words we need to learn, the phaseshift of a sine wave is the horizontal distance that the wave is shifted from the traditional starting position.0254
Let me rewrite the equation here, asin(bx+c)+d.0269
The traditional starting position for sine would be at (0,0), and the traditional starting position for cosine would be at (0,1).0280
Those are the traditional starting positions but the phaseshift will move the graph to the right or the left.0302
In these equations, it's given by c/b, and that seems a little mysterious and let me explain that a little bit.0311
We can write this bx+c, first of all, we can factor b out, we can write that b[x + (c/b)].0320
Then, we can write that as b[x  (c/b)].0332
That's where that c/b comes from.0340
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.0342
Let me draw that, I'll draw that in blue.0367
If you're starting with a sine curve, the phaseshift is the amount that it moves over, that's the phaseshift right there, it moves it over c/b units.0370
If you're starting with a cosine curve, that's that phaseshift right there.0393
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.0407
Let me start with a sine curve.0426
If we apply a vertical shift to that, that amount right there, I'll draw this in red, that's the vertical shift.0434
That's a little bit easier to pick out than some of the others, because that's just the d in the original equation.0455
If you're moving the graph up or down by an amount of d.0464
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.0469
It's a little bit tricky but we'll go through some of the examples and you'll get the hang of it.0479
In our first example here, we're given an equation 3cos(4x+π)+2.0485
We have to identify all these various things, the amplitude, the period, the phaseshift and the vertical shift and then we're going to draw a graph of the function.0495
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.0502
The graph isn't too hard as long as you these things in order.0510
The amplitude, remember, that's just the number on the outside, the a in the original equation.0517
Let me write down the original equation, acos(bx+c)+d.0528
The amplitude is just the a right there, that's the 3.0539
The period is 2π/b, the b there is 4, that's 2π/4 which is π/2.0543
The phaseshift is c/b, our c here is π, π, b is 4, that's π/4.0563
The vertical shift is just that last term d, which is 2.0580
Those are the answers to the first part of the question.0596
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.0602
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.0611
Then move it around according to each one of these parameters.0620
The key thing here is you have to do it in order, you have to do amplitude period phaseshift then vertical shift.0624
Let's see how that works out.0630
Let me draw a basic cosine graph and then we'll try moving it around according to these different parameters.0634
Remember, basic cosine graph, there's π, there's 2π, π/2, 3π/2.0645
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.0661
That's the basic cosine graph, you pretty much have to remember that to get started here.0676
First thing we're going to do, is we're going to change the amplitude.0682
Let me keep track of this as I go along.0688
First one we graphed was cos(x), just y=cos(x).0692
Next we're going to graph is 3cos(x), we're going to bring in the amplitude.0697
What that does is that it stretches up the peak, then it stretches down the valleys by a factor of 3.0700
I'm going to draw the same shift graph but three times as tall, and three times as deep.0709
Instead of going from 1 to 1, goes down to 3 and up to 3.0720
That second graph I drew there was 3cos(x).0725
The next one is to introduce is the period.0734
The period is supposed to be π/2, remember, the period is the amount of horizontal distance between one peak and the next peak.0738
My current peak is...0747
I'm going to change the period to π/2.0754
What I'm really graphing here is 3cos4x.0760
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/2.0769
There's π/2 right there.0784
I need to do a whole period between there and there.0787
Every π/2, it does a complete cycle.0800
What I just drew was 3cos4x.0811
The phaseshift 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.0817
I'm going to see if I can draw this in red or will see if it's still visible.0827
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/4.0832
That's the graph in red there, 3cos4x+π because I've introduced the phaseshift in there.0852
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.0860
We had π/2, π, π/2.0876
I'll redraw the red one on this set of axis, π/2, and that one is going from 3 to 3.0883
Remember the red one is the previous graph shifted over by π/4.0911
Finally, we need to introduce the vertical shift of 2, and I'll do this last graph in blue.0916
That takes the entire graph and raises it up by 2 units.0922
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 1.0927
Let me label that more clearly, 1.0942
I'll draw this one in blue, this is now 3cos4x+π, and I'm introducing the vertical shift of +2.0947
I'm taking the graph and I'm moving it up 2 units there.0965
That blue graph at the end is our final function.0972
This is really a pretty complicated process.0980
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.0983
Let me just recap there, we started with the original graph of cos(x), that's the starting point.0993
Then we introduced the amplitude, and that stretches it vertically by a factor of 3, stretches it up and down.1000
We introduced the period which compresses it horizontally.1013
We introduced the phaseshift, which takes the whole thing and it moves it to the right or the left.1020
Finally, We introduced the vertical shift which takes the whole thing and moves it up or down.1030
Remember, it's important to do these in order, amplitude, period, phaseshift, vertical shift.1041
If you do those out of order, then they'll mess each other up as you go along.1050
We really want to do those in order.1055
We want to practice several of these, so let's get moving in other example.1059
Here's another example, same questions here, amplitude, period, phaseshift, and vertical shift of the following function.1061
Remember we can read this off quickly, just remembering the formula asin(bx+c)+d.1070
If we can figure out what a, b, c, and d are, we have formulas for all of these properties.1080
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.1089
The period is 2π/b, the b is 2 here, so that's π.1098
The phaseshift is c/b, which is, okay, c is π/3, so negative of that is (π/3)/2 will give us π/6.1109
Finally, the vertical shift is d, which ,in this case, is just 0.1135
Finally, the fun part, we get to graph the function1154
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.1159
Let me start with this one's a sine graph.1169
I know the basic shape of the sine graph, I've got that memorized, π and 2π.1174
I know sine always starts at 0, goes up to 1, comes back down to 0, to 1, and then back to 0.1187
There's 1, 1, there's π, π/2, 3π/2, so that's my basic sine graph.1199
That's the first thing I graphed there, sin(x), remember this is the graph of sine and cosine x.1209
Now, we start introducing these other attributes and it's important to go in order.1217
First of all, we're going to introduce the amplitude.1222
The amplitude is 2, but we're really multiplying the graph by 2, 2sin(x) is what I'm going to graph next.1225
2sin(x) that stretches it vertically because of the 2, but it also flips it vertically.1235
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 0.1246
That's first one was a little lopsided, let me just see if I can make that a little more, a little smoother.1270
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.1280
Next thing I'll introduce is the period.1288
The period is supposed to be π, so I'm graphing 2sin(2x).1292
That speeds the whole thing up, it shortens the period because the period is now π instead of 2π.1298
I need to do that entire graph in the space of π instead of 2π.1306
There, that one that I just graphed, I shortened the period to be π instead of 2π.1330
The period is now π on that new graph.1337
Next, we're going to do the phaseshift, that's π/6 units to the right.1340
The phaseshift, 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.1349
Let me do this one in red.1364
I'm going to take that last graph and shift it over π/3 units to the right.1370
Instead of starting at (0,0), it starts at π/3, it's going to come back down to 0, at 5π/6.1374
The phaseshift is supposed to be π/6, so I'm going to move everything over by π/6.1389
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/3.1408
That red curve that I graphed there is 2sin(2x)π/3.1439
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.1453
This last graph is the one we want.1464
Again, it's a matter of breaking these equations down into their parts.1470
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.1475
Remember the equations for amplitude, period, phaseshift, and vertical shift.1484
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.1489
You change the period which compreseses it horizontally. 1500
You do the phaseshift which takes the whole thing and without compressing it, it moves it to the right or the left.1507
Finally, the vertical shift moves it up or down.1511
You could just keep moving these graphs around until you build up the equation you're looking for.1515
This one's a little bit different, we're asked to find the sine wave.1526
This time we're told what all the properties are.1530
We're given the amplitude, the period, the phaseshift, and the vertical shift.1535
We want to find an equation and we want to graph the function.1540
I'm going to kind of build this up, the same way we're building the earlier graphs.1543
I'm going to start with the basic sin(x), so that's my basic sine wave.1548
Now, I'm going to give an amplitude 2, and remember 2 is just a number on the outside, the a.1554
Let me rewrite that equation, asin(bx+c)+d.1559
Amplitude 2 means a is 2, 2sin(x).1566
Now, period 4π, remember our equation for period was 2π/b, that is supposed to be equal to 4π.1575
When you solve that out, that tells us that b=1/2, so the b has to be equal to 1/2.1597
That means our equation is now 2sin(1/2)x, so we've incorporated the period.1610
Phaseshift is supposed to be π/2, but remember the phaseshift, our formula for that is c/b.1622
b is already, we figured out as 1/2, so that's c/(1/2).1643
If we do a little bit of algebra here, we get (1/2)π=2c, so c=π/4.1650
I got that from the equation c/b=π/2.1667
I already figured out my b, now I can figure out my c.1672
The next part of that equation is 2sin[(1/2)x(π/4)].1677
Finally, I want to talk about the vertical shift, which is supposed to be 1 and that's the d.1692
Finally, my equation is 2sin[(1/2)x(π/4)]+1.1704
That is the sine wave that I'm looking for.1715
Now, I want to graph that thing.1722
I start out with the basic sine wave, π, 2π, π/2, 3π/2.1730
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.1742
Now, I'll introduce these properties in order.1753
I'll start out with the amplitude.1755
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 2.1760
It'll stretch the thing up, and down.1774
Period is 4π, that means the thing stretches out, so that it'll only does one cycle every 4π.1783
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.1794
Now, phaseshift π/2, that means the thing is going to shift π/2 units to the right.1817
I better draw a new set of axis here.1824
I've got π, 2π, 3π and 4π, 5π.1835
What I want to do is take the graph I have above and move it π/2 units to the right because we have phaseshift π/2.1850
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.1866
So, connect these up.1887
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.1913
That means instead of going or peak at 2, it's going to peak at 3 now.1920
Instead of going down to 2, it gets moved up by 1 unit so it's going to go down to 1.1932
Let me draw this last final curve in red.1940
Everything gets moved up by 1 unit.1948
I'm going to plot some points here, moving everything up by 1 unit.1953
That final curve is the one we want.1976
That's 2sin[(1/2)x(π/4)]+1.1980
Again, it's a complicated procedure but if you take it step by step, each one of the steps is not too hard.1991
First, we kind of reconstructed the equation from these parameters that we were given.1997
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.2004
Then, we went through step by step.2020
We started with the basic sine curve.2024
We changed its amplitude, stretched it vertically.2030
We changed its period which stretches out horizontally.2034
We changed its phase shift which moved it over horizontally not stretching but just moving it without stretching.2036
Then, we did the vertical shift, moving it up or down.2041
You should practice a few of these curves on your own.2045
We'll come back and try some more examples together later.2047
1 answer
Last reply by: Dr. William Murray
Wed Mar 18, 2015 8:10 PM
Post by Napolean Richard on March 16, 2015
Dear sir,
As the amplitude can the value of A.SO if there are two possible answers in the Example 3:2sin(pi/2*xpi/4)+1 and 2sin(pi/2*xpi/4)+1?and so does the graph?
1 answer
Last reply by: Dr. William Murray
Sun Jan 4, 2015 7:35 PM
Post by Rasheed Abdullah on December 31, 2014
Thank you for the wonderful lecture. However I noticed that, because the amplitude is always the absolute value of "A," the number there can be either negative or positive and still form the same graph. Is this true for all sine and cosine waves or is there a change to the graph when a negative number is used?
Thank you,
Rasheed A.
1 answer
Last reply by: Dr. William Murray
Mon Nov 3, 2014 9:16 PM
Post by Carroll Fields on November 1, 2014
I have another question. Why on the period portion of the question "B", are you writing for example, in Extra Example II: 2/3x, I thought it would be 2/3pi.
Thanks a lot for the lecture, it helped me very much in learning this concept.
Rusty
1 answer
Last reply by: Dr. William Murray
Mon Nov 3, 2014 9:15 PM
Post by Carroll Fields on November 1, 2014
Can you please explain again the math behind the vertical shift: C/B.
How you factored out B?
Thank You,
Rusty
1 answer
Last reply by: Dr. William Murray
Tue Aug 5, 2014 3:05 PM
Post by Jason Wilson on July 21, 2014
My previous question is referencing example one time around 13:5014:50ish thnks
1 answer
Last reply by: Dr. William Murray
Tue Aug 5, 2014 3:06 PM
Post by Jason Wilson on July 21, 2014
a couple of Quick questions. The phase shift of course takes on the transformation properties.  goes right, positive goes left, am I correct on that thought?
If so the formula for phase shift contains c/b. If both c and b are positive in the problem, and then you plug those positive values into the formula, they then turn negative because of the formula, then why isnt the
phase being shifted right , hence the negative sign in the formula?
1 answer
Last reply by: Dr. William Murray
Thu Apr 3, 2014 1:52 PM
Post by Christopher Lee on March 29, 2014
In Example 3, couldn't A = 2, since the amplitude is defined as A, abs(A)? abs(A)=2 means that A can equal 2 or 2, right?
1 answer
Last reply by: Dr. William Murray
Sat Feb 1, 2014 12:19 AM
Post by Mae Linda Vidal on January 22, 2014
Finally I understand how to do this! Thank you so much! Does it matter which one to use (sine or cos) if given a graph and needing to find the equation for it?
1 answer
Last reply by: Dr. William Murray
Fri Nov 22, 2013 5:44 PM
Post by Carolyn Lesperance on November 21, 2013
If you are given the graph of a sine function and need to write an equation for it, how would you determine the horizontal stretch/compression?
5 answers
Last reply by: Carroll Fields
Wed Feb 26, 2014 3:10 PM
Post by Suhani Pant on August 2, 2013
In Extra Example II, can you explain the math involved in stretching out the period to 3pi?
2 answers
Last reply by: Jason Wilson
Mon Jul 21, 2014 6:29 AM
Post by Monis Mirza on May 17, 2013
Hi,
For the phase shift, how would we know if the graph moves to the left?
1 answer
Last reply by: Dr. William Murray
Wed May 22, 2013 3:14 PM
Post by Jonathan Traynor on May 17, 2013
May I commend and congratulate you on doing such an incredible job on explaining what I previously found such a difficult concept. In example 2 we had an amplitude of 2. Could you please explain why a minus amplitude causes the curve to flip. Thanks again and I am finally understanding and enjoying Trig. You are a great teacher!!!!!
1 answer
Last reply by: Dr. William Murray
Sun Apr 28, 2013 10:24 AM
Post by varsha sharma on April 27, 2013
On asine(bx+c)+d
Doesn't the wave shift left since phase shift is negative c/b ?
1 answer
Last reply by: Dr. William Murray
Tue Apr 16, 2013 8:25 PM
Post by Dave Seale on April 13, 2013
on EX2 2sin(2xpi/3) the phase shift graphed in red pen shows the left side of the function intersecting the yaxis at 2 units instead of crossing the yaxis at 1 unit which is where the function would lie after the phase shift and remain since the vertical shift is zero. Honest illustration error, but I didn't want anyone to be confused!
1 answer
Last reply by: Dr. William Murray
Mon Oct 1, 2012 5:10 PM
Post by Jialan Wang on September 28, 2012
you should have some practice after every lecture
otherwise, you can't practice what you have learned in the lecture
1 answer
Last reply by: Dr. William Murray
Fri Aug 31, 2012 5:32 PM
Post by Timothy Ellis on July 27, 2012
why bother going throught the whole process of reversing the equation when you just end up graphing what was given in the example to begin with?
1 answer
Last reply by: Dr. William Murray
Fri Aug 31, 2012 5:36 PM
Post by Timothy Ellis on July 27, 2012
having a hard time figuring out how the phase shift becomes pi/4. I can't seem to see the algebra could you break it down for me please?
1 answer
Last reply by: Dr. William Murray
Fri Aug 31, 2012 5:45 PM
Post by Safreeca Logan on June 13, 2012
your a great instructor. I've learned more fro you in a couple days thsn this whole semester
1 answer
Last reply by: Dr. William Murray
Fri May 18, 2012 1:15 AM
Post by Michael Feldman on April 3, 2012
the quick notes don't match the video
1 answer
Last reply by: Dr. William Murray
Tue Apr 16, 2013 8:16 PM
Post by dorian pedraja on February 27, 2012
at the 13 minute mark, why is the period (pi/2) = 3cos(4x)? that makes no sense to me
1 answer
Last reply by: Dr. William Murray
Tue Apr 16, 2013 8:11 PM
Post by Irene Holly on January 18, 2012
Exsmple 3 should have been pi/4 not pi/2 if i remember correctly... Under Amp and period of a sin wave?
5 answers
Last reply by: Dr. William Murray
Tue Apr 16, 2013 8:04 PM
Post by Robert Haycock on May 14, 2011
H
1 answer
Last reply by: Dr. William Murray
Tue Apr 16, 2013 8:03 PM
Post by Mike Jones on March 29, 2011
question at 7:00 minutes  should that be "B(X + C/B)" becomes "B[X(C/B)]", NOT becomes "B[X(C/B)" ?