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The sine function is generally written like this:
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F(x) = sin X
 
 and makes a graph that looks like this:
 We can change this function in two different ways.
 We can do things with the X.
 For example, we could change the X to 3X to get:
 
F(x) = sin (3X)
 
 We could also change the whole function.
 For example, we could multiply it by 3 and subtract 2 from it:
 
F(x) = 3(sin X) - 2
 
 Each type of change we do will do a specific kind of thing to the graph.
 
 If we subtract 2 from the sin X, the whole graph will move down 2 units:
 
 F(X) = sinX
 
 F(X) = sinX - 2
 
If we add 3 to the sinX, the whole graph moves up 3 units:
F(X) = sinX + 3

 
 Putting a number in front of the sinX
 means we multiply the output by that number.
 If the number is bigger than 1, the graph is stretched up and down.
 
 If we put a 2 in front of the sinX,
 the graph will go up to 2 instead of 1 and down to -2 instead of -1:
 

 F(X) = 2 sinX

  
 If the number in front of the sinX is less than 1, the graph will be squashed
 
 F(X) = 1/2 sinX
 
 If the number in front of the sinX is negative,
 the whole graph is flipped over top to bottom.
 If the negative number is a bigger number than 1,
 the graph will also be stretched.
 
 F(X)= -2 sinX
 
 F(X) = - 1/2 sinX
 
 That's about all we're going to do on the outside of the sinX.
 Now let's talk about how the graph changes
 when we change the X to something more complex.
 
 The things we can do to the X are the same things we did to the whole sinX.
 
 We can add or subtract something. Like say: F(X) = sin(X+90°)
 We can multiply the X by something. Like say F(X) = sin3X
 
 If we add something to the X, the graph moves to the left.
 
 F(X) = sin(X+90°)
 
 If we subtract something from the X, the graph moves to the right:
 
 F(X) = sin(X-90°)
 
 OK, here's a pop quiz.
 Look at the last two graphs.
 These were graphs of the sine function that had been changed in some way.
 They can also be thought of as graphs of the cosine function
 that may have been changed in some way.
 The question is, what would you have to do to the cosine function
 to get the two graphs above???
 
 ANSWERS:
 To get the first one, you don't need to do ANYTHING.
 It is the graph of the cosine function.
 To get the second one, you need to flip the cosine function over. 
 That is F(x) = -1 x cosX or just F(x) = -cosX
 
 That means:

sin(X+90°) = cosX    and    sin(X-90°) = -cosX

 
 OK, that's no big thing, 
 but we will be doing a bunch more of that kind of thing later on.
 
 The last thing we need to talk about is a number next to the X. 
 That is, multiplying the X by something.
 
 If the number is greater than 1, 
 the graph is squashed in from the left and right like squeezing a spring.
  
 F(X) = sin(2X)     ( you can also write this just:  F(x) = sin2X )
 

 
 If the number is greater than 0 and less than 1
 the graph is stretched out to the left and right.
 
 

 If the number next to the X is negative, the entire graph is flipped left to right. 

 The graph will also be stretched if the number is between 0 and -1.
 The graph will also be squeezed if the number is less than a -1:
 
 F(X) = sin(- 1/2 X)

 
 F(X) = sin(-2X)

 
 You are allowed to do as many of these things at the same time as you want.
 You could have:

F(X) = 2sin(4X) + 1

 
 The 2 stretches the graph up and down to + 2 and -2.
 The 4 next to the X squeezes the graph from left to right
 so it completes a cycle in 90° rather than 360°.
 
 The +1 at the end moves the whole graph up 1 unit so it goes from +3 to -1
 
 When you graph it, it looks something like this:
 Let's recap:
 
 These changes work the same way for all of the trig functions.
 We could just as easily write:
 
F(X) = Acsc(BX - C) + D
or
F(X) = Atan(BX - C) + D
 
copyright 2008 Bruce Kirkpatrick