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Investigating Sin Functions

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Introduction

Adam Hussain IB Math SL - Internal Assessment (Practice) Part 1 - (Y = A sin(x)) In this assignment, I will investigate graphs of Sine Functions. The graph below depicts the basic, unchanged Sine Function, using graphing software from www.FooPlot.com to graph. Y = Sin(x) As we observe the basic Sine function, we can notice significant differences when compared to the graphs y=2sinx, y= (1/3) sin x, and y= 5sinx. First, let us compare the graphs y= (1/3) sin x, and y=sin x, respectively. Y= (1/3) sin (x) y= sin (x) As we can see from comparing y = (1/3) sin x and y = sin x, the first respective graph has been shrunken vertically from the original sin x graph. (In other words, the Y values of the graph have been shrunken by (1/3). Let's see what occurs when we use FooPlot to graph a Sin function with even lower A, or Amplitude. Y = sin(x) Y= (1/4) sin(x) Y= (1/8) sin(x) As we examine these three graphs with decreasing "A" (amplitude) values, we can plainly see the graphs are shrinking vertically as "A" decreases. Now, let's compare what happens when we decide to increase the amplitude, or A, when we enter the equation into the graphing software on FooPlot. ...read more.

Middle

If A>1, the graph will stretch vertically, and if A<1, the graph will shrink vertically. Part 2 - (Y = sin Bx) In sinusoidal functions, B represents the number of cycles a graph completes in an interval of 0 to 2???This value B affects the period, which is the time interval for the function to happen, after which the function repeats itself. Graphically, the period is the length of one full cycle of the wave. To see how B value affects the graph, let's graph it with FooPlot: Y = Sin (x) Y = Sin (2x) As we can see from the graphs, Sin (2x) is a compressed version of the original function. This new function has a shorter period because it now takes a shorter amount of time to finish a cycle. But by how much has it been compressed? We can determine this algebraically. Because the new period of a graph is determined by the equation 2??|b|, and the B value in this case is 2, the new period is 2?????r??. Now, we can determine algebraically how to get from 2??to ??a different way. If we divide ? by ??? we get 1/2. And this is the same as saying 1/|b|. Therefore, the graph changes by a factor of 1/|b|. ...read more.

Conclusion

My graph confirms this: Finally, let's look at -sin (1/2 (x-1)). This graph will be reflected across the x axis, horizontally stretched by a factor of 2, and horizontally translated to the right of the origin by one unit. My graph confirms this: Using my conjectures, I can use the Amplitude of the equation to predict if it's horizontally stretched or shrunk, B to determine the period, and C to determine the horizontal translation. Part 5 - Cosine Function How is the graph of y= cos x linked to the graph y =sin x. What's the relationship? Well let's start by comparing the graphs side by side using FooPlot. For the sake of comparison, I will use a different grapher to display the exact local extrema of the graphs. <--This is y=sin(x), with a maximum at x= ??2 <-- This is y=cos(x) <-- This is the overlap of both graphs. According to the graph, in order to the get the Cos (x) curve from sin (x), we need to shift the sin curve by ??2 to the left. Therefore, cos x = sin (x+ ??2) And, if we refer back to the graph, it follows logically that to get the sine curve, we should shift the Cosine graph over ??2 to the right. Therefore, sin x = cos (x- ??2) ...read more.

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