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# Mathematical Investigation

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Introduction

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|.

Middle

Part III

Investigate the family of curves y=sin(x+c)

Figure#5: Graphs of different functions of the family curve y=sin(x+c):y=sin(x); y=sin[x+ (pi/4)] and y=sin[x-(pi/4)]. According to Figure #5, a change in the value of “c” shifts the entire function to the right or left of the origin; this is called a phase shift. When the red graph (y=sin(x) original) is compared to the blue graph [y=sin(x+ (pi/4)), the blue graph shifted 1/4 pi to the left while the length of each period and the amplitude remained the same with y=sin(x). When the red graph (original y=sin(x)) is compared to the green graph [y=sin(x-(pi/4)], the green graph is shifted 1/4 pi radians to the right while the length of each period and the amplitude remained the same with y=sin(x).

Conjectures

(a)

Based upon the observation above the transformation of the basic function f(x) = a∙sin (bx+c) +d with varying values of “c” is a horizontal translation along the x-axis; the direction of the translation is dictated by the zero of “c” in the angle (bx+c). For instance the zero of (bx+c) is –c, which means a horizontal translation along the x-axis to the left; vise versa if the zero of the bracket produces +c, the translation would be towards the right. The value of this horizontal translation is the vale of “c”.

Conclusion According to Figure #11, the standard cosine graph y=cos(x) is a horizontally translated of the standard sine graph y=sin(x) to the left by pi/2 radians. To test this hypothesis the standard cosine graph is shifted pi/2 radians to the right in the equation y=cos(x-(pi/2)) and this graph is compared to the standard sine graph. If y=cos(x) is nothing but a horizontal translation of pi/2 radians to the left of y=sin(x), then sin(x) =cos(x-(pi/2)).

Figure #12: Graphs of the function y=cos(x-(pi/2)) and y=sin(x) According to Figure #12, the standard sine graph y=sin(x) = a horizontally translated to the right by pi/2 radians of the standard cosine graph y=cos(x). Based upon the Figure #11 and Figure #12, I conclude that cosine graph has the same amplitude and period/complete cycle as the standard sine graph; cosine is horizontally translated pi/2 radians to the left in reference to y=sin(x).

Based upon the properties observed two relations are obtained:

sin (x)=cos(x-(pi/2))

cos (x)=sin(x+(pi/2))

By combining the properties of the reflection on the x-axis and translation by the standard sine function y=sin(x), the standard cosine function y=cos(x) can also be obtained.

Figure #13: Graph of y=-sin(x-(pi/2)) and y=cos(x) According to Figure # 13, the reflection on the x-axis with horizontal translation to right (by pi/2) by the sine function is the same as y=cos(x). Therefore based upon the properties gathered from Figure #13, two relations are obtained:

cos (x)=-sin(x-(pi/2))

sin (x)=-cos(x+(pi/2))

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

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