# Mathematical Investigation

Type I: Mathematical Investigation

The equation of the sine function is defined as f(x) = a∙sin∙b(x+c) +d; f(x) is the function notation; a is the amplitude; sin is the function; b is the representation of the stretch factor; c represent any phase shift; d is the y intercept. (Any transformation of the sine function is compared to y=sin(x)).

Part I

Figure #1: The graph of y=sin(x)

Figure #2: Comparison of different equations of sine functions: y=sin(x); y=2sin(x); y= (1/3) sin(x); y=5sin(x).

Figure #3: Investigation of other sine graphs

Based upon Figure #1 to Figure #3, as “a” varies only the amplitude of the graph changes according to |a|. Since only the value of “a” changed, there is no change in the period, no phase shift, no change in X and Y intercepts. As these figures illustrated, every single graph plotted has the same period (period is the horizontal distance between two points on the x-axis; period usually defined as the horizontal distance along x-axis of one complete cycle, 2pi radians) and the same X and Y intersects. There is no phase shift in any of the graphs because the position of the periods and x intercepts of each graph is the same.

Conjectures:

(a)

The transformation of the standard sine function y= a∙sin(x) by varying values of “a” only changes the standard sine function y= a∙sin(x)’s amplitude according to |a|. Varying values of “a” does not change the function’s amplitude or X and Y intersects. When the value of “a” increases the function is stretched vertically and so the amplitude increases as well. Vise versa, when the value of “a” decreases, the function is vertically compressed and so the amplitude decreases as well.

(b)

Based upon the properties observed from above figures, varying values of “a” appear only change the height (amplitude) of wave pattern and nothing else. Different values of “a” does not effect the period as the period in all graphs remain the same. Based on the observation that varying values of “a” only changes the amplitude, therefore it can be easily conjectured that the waves are stretched vertically, while the X-intercepts remain the same, according to |a|.

Part II

Investigate the graphs of y=sin (b∙x)

Figure #4: Comparison of graphs of y=sin(x); y=sin (2x) and y=sin (4x).

According to Figure#4, as the value of “b” increases from 1 to 2, as indicated by the red graph [y=sin(x)] and blue graph [y=sin(2x)], one complete cycle of the blue graph (1pi radians) only occupies ½ of the period of the red graph (2pi radians). Similarly, when compared the red graph [y=sin(x)] to the green ...