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Graphing Sine and Cosine


Date: 07/07/2000 at 13:56:36
From: Bernadette
Subject: Graphing sine and cosine

I just don't understand how my teacher gets the points that he uses on 
the graph. I have tried to do the same thing he does over and over 
again, but it doesn't work. Please help me.


Date: 07/07/2000 at 14:34:56
From: Doctor Jaffee
Subject: Re: Graphing sine and cosine

Hi Bernadette,

I assume that you are talking about sketching the graph of a sinusoid; 
that is, an equation in one of the following forms:

     1. y = C + AsinB(x - D)
     2. y = C + AcosB(x - D)

Here is how I would draw the graph.

First, draw a dotted horizontal line C units above (or below if C is 
negative) the x-axis. This dotted line is the transverse axis. Half of 
the graph will be above the dotted line and half will be below.

If D is positive, go D units to the right of the origin and, if you 
are drawing a type 1 equation (the one with the sine), go up to the 
transverse axis and put a dot. This will be the first point of the 
cycle. If D is negative, go to the left.

If your equation is type 2, go D units to the left or right just as 
I suggested for type 1. Then if A is positive, put the dot A units 
above the transverse axis. If A is negative, the point will be below 
the transverse axis.

Now, if you are doing a type 1 graph and x is in degrees, then the end 
of the cycle will be on the transverse axis, 360/B degrees to the 
right of the first point of the cycle. Halfway between these two 
points on the transverse axis will be the midway point of the cycle. 
If the equation is in radians, then the last point will be 2*pi/B 
units to the right of the first point.

Halfway between the first point of the cycle and the midway point of 
the cycle, draw a point A units above (or below, if A is negative) the 
transverse axis.

Halfway between the midway point and the end of the cycle, draw a 
point A units below (or above, if A is negative) the transverse axis.

By now you should have 5 points. If you draw a smooth curve connecting 
them, you will have one cycle of the sinusoid.

If you are graphing a type 2 sinusoid, then the first and last points 
of the cycle will be A units above the transverse axis and the midway 
point will be A units below the axis. Of course, if A is negative, 
just the opposite will happen. The points 1/4 and 3/4 of the way 
through the cycle will be on the transverse axis.

I hope this explanation helps. Try a few examples and see what 
happens. If you are having difficulties or if some of my explanation 
was unclear, write back and I'll try to help you some more.

- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Equations, Graphs, Translations
High School Trigonometry

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