EMAT 6680

Assignment 1

Kathy Radford

Fall 2008

 

An investigation of the effect of a, b, and c on the graph of y=a sin (bx+c).

 

 

1)          For our first step we will take a look at the graph of a basic sine function, by setting a = b = 1 and c equal to 0.

 

y = sin x

 

:::SineWave1.gif

 

 

 

1)          Now see what happens when we vary a from 0 to 5:

 

y = a sin x,  0 < a < 5

 

 

 

 

2)         We can also make a negative, and by comparing it to the same function using abs(a), we can see that the negative a reflects the graph across the x axis.

 

y=-1 sin x (in red), and y = abs(-1) sin x (in purple)

 

 

:::SineNeg.gif

 

3)         When a = 1, the range of a sine function is -1 to 1.  Using a value of a other than 1 changes the amplitude of the graph.  Amplitude modulation, which is what the AM stands for in AM radio, uses a lower frequency audio signal to modulate a higher fixed frequency signal for wireless transmission and other applications.  For example, the following function displays amplitude modulation replacing a with a 3 hz cosine function and using it to vary the amplitude of a 100 hz sine function.

 

y= cos(6 pi x)sin(200 pi x)

 

 

:::SineMod.gif

 

 

 

 

Part 2

 

 

  Now lets see what happens when we vary the b in y=a sin(bx+c).  Setting a =  1 and c= 0 we have the equation y=sin(bx).  When we set b equal to 1 we can see that the sine function goes through one complete cycle as x progresses from 0 to 2p, or approximately 6.14, cycling from 0 to a maximum value of 1, then back through 0 to a minimum of -1 and returning to 0. This is shown in the graph as the function in red.

 

When the value of b is doubled to 2, it can be seen that the function goes through two cycles in the same period of 2p.  This is shown in blue.

Red:   y=sin(x)

Blue:  y=sin(2x)

 

 

 

The period can also be lengthened by using a value less than one for b, such as 2/3, shown below in purple.

 

y = sin(2x/3)

 

 

 

 

 

 

And, when we make b negative, the graph of the function is reflected across the x axis, as shown in green below.

 

y=sin(-x)

 

 

 

 

 

 

By increasing b to a much larger number, it can be seen that the function will go through one cycle much sooner.   A close inspection will show that the interval in x required to go through one cycle equals 2p/b.

 

 

 

 

 

 

Part 3

 

The final consideration for this assignment is the effect of adding a constant to x in the equation.  We can try this out by adding 2 to x.

 

y= sin(x+2)  (purple),    and y=sin(x)    (red)

 

 

 

This has the effect of shifting the function to the left by 2, which is similar to the horizontal transformations performed on other functions.  We would expect that  a negative number would shift it to the right:  here is y = sin(x – 1)

 

 

 

 

And, a shift of 2p should shift the function one entire period, and appear identical to the original function.

 

y = sin(x + 2p)