Exploration of the Sine Curve
Examine the graphs of y = a sin (bx+c) for different values of a, b, and c. To make it easier for the reader, all graphs will include y=sin(x) in grey as a standard.
Examination when we vary “a”:
As the graph shows, when “a” is altered, the amplitude of the graph changes. In essence, the crest of the sine wave is at the given value of "a". If “a” is negative, as the blue line shows, the crest becomes a trough at the value of “a”.
The case where “a” is a fraction (less than 1 and greater than -1) was examined too, but the results support the prior conclusion: the crest of the new sine wave is at “a”, regardless if “a” is positive or negative.
Examination when we vary “b”:
When “b” is altered, the period of the graph changes. The higher the value of “b”, the smaller the period becomes (when b is greater than 1 and less than -1). When “b” is negative, the graph is reflected over the x-axis. Note that two graphs have been shown using the same equations to illustrate the smaller periods.
After examining this, I wondered what would happen if b was a fraction less than 1 and greater than -1. Hence, the next graph below shows my further exploration. Upon construction, I could see that fractions (less than 1 and greater than -1) increase the period. Again, when the fractional “b” is negative, the graph is reflected over the x-axis.
Examination when we vary “c”:
When “c” is altered, the graph shifts along the x-axis. When “c” is positive, the graph is moved to the right. The higher the value of “c”, the further the graph is moved. When “c” is negative, the graph is moved to the left.
In this examination, having a fractional “c” that is less than 1 and greater than -1 does effect the results. When “c” is greater than 0 and less than 1, the graph is shifted to the left along the x-axis. The larger the fraction, the further the graph is shifted. When “c” is less than 0 and greater than -1, the graph is shifted to the right along the x-axis. Again, the larger the fraction, the further the graph is shifted.
