Skowhegan Area High School
 TOPICS Introduction Linear Exponential Power/Inverse Power Periodic Square Root Absolute Value
 Math 3 Unit VI Periodic Functions Period Amplitude Center line Stretch Shift Periodic Function y=sin(x) y=cos(x) Periodic functions are functions that repeat after a period of time, such as sine and cosine functions. The sine function passes through the origin in its original format, splitting the wave half way between the max and min of the wave. The cosine is has its maximum portion of the wave on the y axis where x=o. Reflection (x) The reflection is done by making the entire function negative, this makes the function reflect over the x axis. When reflected across the x axis each y value becomes the opposite value. The function is now written as y=(-)sin(x) or y=(-)cos(x). Period The period is the length of time it takes before the function will repeat again. For y=sin(x) or y=cos(x) (the parent functions) the period will be 2 pie (360 degrees). Amplitude The amplitude is the distance from half of the wave to the top of the wave (half way between the minimum point and maximum point is half the wave). For the parent function y=sin(x) the amplitude is 1. Stretch Horizontal To create a horizontal stretch the number would need to be divided from the x value of the parent function. To stretch the function one would write y=sin(x/a) or y=cos(x/a). To compress the function the opposite application would be done, multiplication to the x. This would be written as y=sin(ax) or y=cos(ax). Vertical To do the vertical stretch the multiplication (compress) and division (stretch) need to be applied outside the parenthesis. These functions would look like y=sin(x)/a or y=cos(x)/a for the stretch. For the compression they would be y=sin(x)a or y=cos(x)a. Shift Horizontal To apply a horizontal stretch the number must be added to the x value inside the parenthesis. y=sin(x+a) or y=cos(x+a). In both cases the shift is to the left, the positive moves it left as subtraction will shift the function right. An example of a function shifted right would be y=sin(x-a) or y=cos(x-a). Vertical To apply a vertical shift to a periodic function you add or subtract the amount at the end of the equation (to the y value). The number added will be the amount it is raised on the y axis. The number when subtracted will be the amount it is shifted down the y axis.