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
1) Now see what happens when we vary ÒaÓ from 0 to 5:
2) We can also make ÒaÓ negative, and by comparing it to the same function using abs(a), we can see that the negative ÒaÓ reflects the graph across the x axis.
y=-1 sin x (in red), and y = abs(-1) sin x (in purple)
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)
Now letÕs see what happens when we vary the ÒbÓ in y=a sin(bx+c). Setting a = 1 and c= 0 we have the equation y=sin(bx). When we set ÒbÓ 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 ÒbÓ 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.
The period can also be lengthened by using a value less than one for ÒbÓ, such as 2/3, shown below in purple.
y = sin(2x/3)
And, when we make ÒbÓ negative, the graph of the function is reflected across the x axis, as shown in green below.
By increasing ÒbÓ 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.
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)