Item 3
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The function y=sin(x) is the green line or the one with the lowest amplitude
Two alternate functions include y=2cos(4x)+1, and y=2sin(4x +π/2)+1.
The cos function is depicted in blue and if the red sine function has its A value changed to 2, B=4, C= 1.57 (or π/2) and D= 1 the function will match the shape of the cosine.
These functions were first found by investigating A, B, C and D. Firstly, the amplitude of the function is 2, hence the the A value for both equations was 2. The functions period was determined from the graph to be pi/2, thus following the formula period/2π=B it was calculated that B= 4. D was found to be 1, hence the +1 at the end of the function. For C, the function started as a cos curve due to its peak being located on the y intercept, however it was also found that if a sine curve is moved forward π/2 radians, or 90 degrees, it will start in the same spot. Consequently the cosine function had no value for C , while in the sine function C= π/2 in the other. It is also noted that from this, there are an infinite amount of functions that will match this graph, simply by moving a function forward or backwards π/2.