sine and cosine sliders
This interactive graph allows you to see the effects of changing A, B, h, and k in the equation y = A*cos(B(x-h))+k (and similarly with sine instead of cosine). It is not tied to a particular assignment, but is to be used as a tool for in class demonstration and reference, and an outside of class reference/study-aid. Note the following: 1) The vertical stretch or compression, and reflections over the x-axis, are determined by A. Also, |A| is called the amplitude of the sinusoidal function (a concept that doesn't apply to general functions). It is the amount that the function oscillates above and below its average ("midline") value. 2) The horizontal stretch or compression, and reflections over the y-axis, are determined by B. Remember that the stretch or compression is by the factor 1/B. Because the period or sine and cosine is 2pi, this creates a new period of 2pi/B. 3) The horizontal and vertical shifts are given by h and k, respectively. Note that these are determined after the stretching and compressing. The "h" determines how far horizontally the "start" has moved (this would be a peak for cosine, and in the middle going up for sine). The "k" is added to the base average value of y=0, so it becomes the new average value, or "midline" value. It is halfway between the maximum and minimum values of the function.