



The sine function is generally written like this:




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(X90°) 








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(X90°) =
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

