Section+4.4

**(4.4)** **__Periodic Function__: **

-A Function y=f(t) is __periodic__ if there is a positive number 'c' such that f(t + c) = f(t) for all t's in the domain of f.

-The smallest such 'c' is called the __period__ of the function.

y = sinx (has a period of 2π)

-A __function__ is a __sinusoid__ if it can be written in the form: y = asin[b(x-c)]+d

'a' = __amplitude __- 1/2 the height of the wave (represents the volume- loud/quiet)

'b' = __new period __- old period / IbI

__Frequency __- The number of complete cycles over a unit interval (represents the pitch) Equation: (IbI / old period)

-__Period__ of sin and cos is 2π

'c' = moves graph left or right
 * -__Phase Shift__: **

'd' = moves graph up or down


 * Ex:** Graph y = 1/2 cos(x+11/2)

y = sin [2(x+3)]-1 New Period = 2π / 2 = π
 * Ex:** Graph y = sin (2x+6)-1

y = sin (x+π/2)
 * Ex:** Write the cosine function as a phase shift of the sine function.

New Period = Old Period / IbI π/5 = 2π/b b = 10 y = 6 sin [10(x-2)]
 * Ex:** Construct a sinusoid with period π/5 and amplitude 6 that goes through (2,0).
 * ^**the 10 represents that it shrinks by 1/10

2 π/2 = π 64 = 2 π/b b = π/32 y = -10 cos [π/32x]+15
 * Ex:** Graph y = sin 2x
 * ^**It shrinks by 1/2
 * Ex:** Construct a sinusoid that rises from a minimum value of y = 5 at x = 0 to a maximum value of y = 25 at x = 32.

-pg 392 (#3-27) Mult. 3 -pg 393 (#54-66) even
 * __Homework Assignments: __**