Friday we took a look at periodic functions.
A function is periodic with period T if f(x+T)=f(x) for all x such that x and x+T are in the domain of f. The smallest such number T > 0 is called the fundamental period.
We dealt with 3 types of periodic functions: the trigonometric functions of sine, cosine, and tangent (f(x)=sin(x), f(x)=cos(x), and f(x)=tan(x) or sin(x)/cos(x)). Remember to keep your calculator in radian mode.
Sine and Cosine functions are stated in the form y=Asin(Bx+C).
- The A is the amplitude (the vertical minimums and maximums from the center line: |A|)
- Finding the amplitude is simple: just take the absolute value of the maximum or minimum vertical distance from the center line.
- If A<0, the amplitude is inverted – the graph is reflected over the center line
- The B is the period of the function (the distance it takes for a function to repeat itself: 2π/|B|)
- To find the period, divide 2π by the absolute value of B.
- The larger B is, the shorter the period. The smaller B is, the larger the period.
- It is a horizontal stretch which creates “b” periods over the original domain.
- The C is the phase shift of the function (the horizontal translation of the function: -C/B)
- To find C, pick a starting point (for instance, if the normal graph – under no translation – would begin at x=0 with y=0, find the first point where y=0). Then, take the negative of that point and divide the number by B.
When the equation resembles y=sin(x) + d, d will translate the graph vertically (it’s the y-intercept; if d > 0, the graph will shift upward, and if d < 0 the graph will shift downward).
Last but not least, y=tan(x)=sin(x)/cos(x). This means, whenever cos(x)=0, the tangent function is undefined.
