By Brooke Norman

Is it just scribble or does it have meaning?

Looking at the graphs of sine and cosine can sometimes just look like wavy lines that have no meaning.  LetÕs take a more in depth look to see what all this scribble is about.

LetÕs take a look at the graph of the equations: y=sin(x) and y= cos(x).  Look to see where they cross the x- and y-axis.

Now letÕs look at the equations of y= a cos(bx+c) and y=a sin(bx+c), and look at different values of a, b, and c.

For each graph, we are going to see what is happening as the graphs cross the x- and y- axis and what the minimum and maximum values for w and y are.  This means, what changes are occurring due to the value changes of a, b, and c.

LetÕs look at the sine graph first.

Here we are going to look at the graph as the value of ÔaÕ is changed, letting b=1 and c=0.

Purple: y=1/2 sin(x)

Red: y= sin(x)

Blue: y= 3 sin(x)

Green: y = 5 sin(x)

Here we are going to look at the graph as the value of ÔbÕ is changed, letting a=1 and c=0.

Purple: y=sin(1/2 x)

Red: y=sin(x)

Blue: y=sin(3x)

Green: y= sin (5x)

Here we are going to look at the graph as the value of ÔcÕ is changed, letting a=1 and b=1.

Purple: y=sin(x)

Red: y=sin(x+1/2)

Blue: y=sin(x+3)

Green: y=sin(x+5)

Now, letÕs look at the cosine graph.

Here we are going to look at the graph as the value of ÔaÕ is changed, letting b=1 and c=0.

Purple: y=1/2 cos(x)

Red: y= cos(x)

Blue:y= 3 cos(x)

Green: y = 5 cos(x)

Here we are going to look at the graph as the value of ÔbÕ is changed, letting a=1 and c=0.

Purple: y=cos(1/2 x)

Red: y=cos(x)

Blue: y=cos(3x)

Green: y= cos (5x)

Here we are going to look at the graph as the value of ÔcÕ is changed, letting a=1 and b=1.

Purple: y=cos(x)

Red: y=cos(x+1/2)

Blue: y=cos(x+3)

Green: y=cos(x+5)

LetÕs combine the graphs of asin(bx+c) and acos(bx+c).

Here is both of them when we vary ÔaÕ

Y=asin(x) and y=acos(x)

Purple:  y=sin(x)

Red: y=3sin(x)

Blue: y=cos(x)

Green: y= 3cos(x)

Here is both of them when we vary ÔbÕ

Y=sin(bx) and y=cos(bx)

Purple: y=sin(x)

Red: y= sin(3x)

Blue: y=cos(x)

Green: y=cos(3x)

Here is both of them when we vary ÔcÕ

Y=sin(x+c) and y=cos(x+c)

Purple: y=sin(x)

Red: y=sin(x+3)

Blue: y= cos(x)

Green: y=cos (x+3)

Here is what all the graphs look like when they are combined?

I guess itÕs more than scribble, huh?