Write up #1 : Investigating the graphs of y=asin(bx+c)
By Raynold Gilles.
A. Introduction
A function that contains the "sin " or "cos" trigonometric ratio is what we commonly refer to as a sinusoidal function. It is most appropriate to events and phenomena that follow a cyclical pattern. In this write up, we will be discussing transformations of tsinusoidal functions in general and The function y = a sin(bx + c) in particular.
Just as one swims in shallow waters before he or she dives deep into high waters; we will first investigate the graph of the parent function " y = sinx"
Click on the link below to see a graph for the function y = sinx
Let us focus on the x values in the interval between -5 and 5.
Notice that the function goes through the origin and reaches its peak at approximately 1.55. ( A value approximately equivalent to half of the number "pi") .
Similary one can observe that the function reaches its minimum at approximately -1.55 ( That is the additive inverse of " half" of the number "pi").
Next, we observe the function for x values on the number line.
*One can notice that the function has a repeating pattern.
*The maximum will be reached at y=1.
* The minimum will be reached at y=2.
Now that we know a little bit about the parent function, we need to familiarize ourselves with some vocabulary terms.
Period
The period of a function is the distance (x value) needed to observe a repeating pattern. In our parent function y = sin x, the period is 2pi or about 6.28.
Amplitude
The amplitude of the function is half of the distance between the maximum and minimum function values. In our case the maximu is 1 and the minimu is -1 for a total distance of 2. Therefore half of the distance will result in 1.
B. Investigation
1. The amplitude.
Now that we have defined the neccessary vocabulary and got somewhat accustomed to the function y = sin (x ); we can investigate its transformations.
Let us look at the function y = 2 sin (x)
Click on the link below to see the graph of the function y=2sin(x)
What happens to the parent function ?
Overall examination of the above graph shows that the amplitude it twice that of the parent function. Hence one can conclude that the value of "a" determines the amplitude of our sinusoidal function.
Nonetheless, one would wonder what a the effect of a negative value of "a" would be. Click on the graph below to compare the graph of y=2sin(x) in Red and y=-2sin(x) in purple.
The graph of y=2 Sin(x) and y=-2sin(x)
One can notice that both the red graph and the purple graph have an amplitude of "2". Hence we can conclude that the amplitude is determined by the absolute value of "a". Also, two functions with values of "a" that are additive inverses of each other are merely a reflection of each other over the x axis.
2. The phase shift.
Next we will compare side by side the graphs of the function y=2sin(x) with that of y=2sin(2x) and that of y=2sin(-2x). Click on the link below to observe graph of the three functions.
The graph of y=2sin(x) , y=2sin(2x) and y=2sin(-2x)
The graphs are as color coded as follow:
Blue graph:y= 2sin(-2x)
Red graph:y= 2sin(2x)
Purple graph:y=2sin(x)
The following conclusions can be made.
*The first two graphs are reflection of each other across the y axis. Therefore, when two values of "b" are additive inverses of each other, they are reflection of each other across they y axis.
*The first and third graph have different periods. Similarly the first and the second graph have different periods. Therefore, one can conclude that the value of "b" also determines the period of a sinusoidal function. Furthemore, the larger the absolute value of "b" is; the smaller the period.
Finally , we will add the constant c to the three graphs above for further analysis. The following functions were graphed.
Yellow:y= 2sin(-2x+5)
Linght blue:y= 2sin(2x-4)
Lime green:y=2sin(x+4)
Our goal will be to compare our graphs in pairs. The following pairs are up for comparison.
Dark Blue vs. Yellow.
Light Blue vs. Red.
Purple vs. Green.
Click on the link below to investigate the effect of the constant "c". One should compare the graphs according to the given pairs above.
Examination leads to the following conclusions.
*Adding a positive constant leads to a horizontal left shift
*Adding a negative constand leads to a horizontal righ shift
*No reflection were observed.
C. Final Conclusion
Both "b" and "c" in our equation affect the phase shift in our function.
The value of "a" on the other hand affects the amplitude.