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Graphing Trig Functions

Date: 09/08/2004 at 09:04:28
From: Ash
Subject: Trigonometry

Hi.  I've been studying how to graph trigonometric functions.  I know
how to find everything.  What I find rather tedious is when it comes
to choosing the x-values.  My teacher taught us to take the period and
divide it by four.  This is our starting point.  Then we add one to
the numerator.  This works well with simple sine or cosine functions.

What about the remaining trigonometric functions?  What if the
function has phase shift?  Regarding the phase shift, our teacher 
suggests to draw the main function w/o the shift, then shift it (as 
if I'm holding the graph and moving it manually along the x-axis).
 
Please I hope you can give me a few guidelines that will make the 
process of graphing trigonometric functions simpler and less tedious.
Thank you in advance.

eg: sin 2x has period 2pi/2 = pi

    x | 0 | pi/4 | 2pi/4 | 3pi/4 | 4pi/4
    ---------------------------------
    y | 0 |  1   |   0   |   -1  | 0

The y-values are based on a pattern.  For any sine function, we'll 
have the same y-values (0,1,0,-1,0) for different x-values obtained 
from the previous method.  It seems a great way but I'm not sure 
whether it's a standard way that works with all the functions or 
it's just a special method for sine and cosines (the pattern for 
cosine is: 1,0,-1,0,1).  If it's special for sines and cosines, how's 
it affected when there is a phase shift?



Date: 09/09/2004 at 17:12:31
From: Doctor Ian
Subject: Re: Trigonometry

Hi Ash,

>Hi.  I've been studying how to graph trigonometric functions.  I know
>how to find everything.  What I find rather tedious is when it 
>comes to choosing the x-values. 

That's for sure.  So perhaps you shouldn't choose them.  :^D

(I'll show you what I mean by that later.) 

>My teacher taught us to take the period and divide it by four.  This
>is our starting point.  Then we add one to the numerator.  This works
>well with simple sine or cosine functions.  What about the remaining
>trigonometric functions?  What if the function has phase shift?
>Regarding the phase shift, our teacher suggests to draw the main
>function w/o the shift, then shift it (as if I'm holding the graph
>and moving it manually along the x-axis). 

That's good advice.  Here's even better advice:  ALWAYS draw the
function the same way, and then just draw the AXES so that they're in
the right place and to the right scale.  

Note that, by this method, you graph sin(x) and cos(x) by starting
with the same curve, and just labeling the origin differently. 

So just draw the curve, which always has the same shape.  If it's a
sine, tentatively assign the origin to be halfway between a trough and
the next crest, where the curve crosses the x-axis.  If it's a cosine,
make the assignment below a crest.  

Now, if it's shifted by some amount, just move the ORIGIN by that
amount.  That is, just start labeling from the shifted location.  If
you need to account for changes in amplitude or period, you can handle
those by changing how you label the graph. 

>Please I hope you can give me a few guidlines that will make the 
>process of graphing trigonometric functions simpler and less tedious.
>Thank you in advance.

Let's look at an example like 

   y = 3sin(2x - pi/4)

     = 3sin(2(x - pi/8))

I'd start by drawing a sine wave, and the x-axis without any points
labeled on it.  It's a sine, so I pick a point between a trough and a
crest.  That's my tentative origin.  I know the whole thing is going
to be shifted by pi/8, so I change the label of that point from '0' to
'pi/8'.  

The period would normally be 2pi, but we've doubled the frequency, so
we should go through one complete cycle by (pi + pi/8).  I move over
by one complete cycle, and label that point '9pi/8'.  And now I've got
the right scale for my x-axis.  I just have to fill in the rest of the
tick marks so that these two points fall where I want them to. 

For my y-axis, I just make a tick mark at the height of the maximum of
the curve, and label it '3', since that's my amplitude.  And now I
have a plot of the function. 

You always want to check with something like this, to make sure you
haven't dropped a sign, or divided when you were supposed to multiply,
or anything like that.  When x is pi/8, the function should be zero:

    y = 3sin(2(pi/8 - pi/8)) = 0       Check.

When x is 9pi/8, it should be zero again:

    y = 3sin(2(9pi/8 - pi/8)) = 0      Check. 

When it's halfway between there (at 5pi/8), it should also be zero:

    y = 3sin(2(5pi/8- pi/8)) = 0       Check.

Halfway to there (at 3pi/8), it should be at a maximum:

    y = 3sin(2(3pi/8- pi/8)) = 3       Check.
  
If it's going through those points, you know it's going to be okay on
the others.  Does that make sense? 

Of course, if you're given labeled axes already, this won't work.  But
that's okay!  What you want to do then is identify the values of x
where the function has a value of 0, or +/-A, where A is the amplitude.  

So given the function above, I'd want to know:  When will it be true that 

   0 = 3sin(2(x - pi/8))

Clearly, at x = pi/8 + k*pi, where k = 1, 2, 3, ...  So I'd go to my
graph and put dots on the x-axis at those points.  Next, I'd want to
know:  When will it be true that 

   3 = 3sin(2(x - pi/8))

This is going to be true when 

  2(x - pi/8) = pi/2

     x - pi/8 = pi/4

            x = pi/4 + pi/8

            x = 3pi/8

plus multiples of pi.  So I can label those points on the line y = 3. 

The point is, if I can find the zero points, and the minima and 
maxima, then I can sketch the rest of the graph easily, since I 
already know the shape. 

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Trigonometry

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