Applications of Periodic Functions

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Suicidal Shield Bug by flickr user ChinchillaVilla

Properties and Transformations of the sine function ...

Let's look at some graphs ...http://fooplot.com

ƒ(x) = AsinB(x - C) + D

ƒ(x) = Asin(Bx - c) + D

D is the sinusoidal axis, average value of the function, or the vertical shift.

The Role of Parameter D

D < 0 the graph shifts down D units.D > 0 the graph shifts up D units.

ƒ(x) = AsinB(x - C) + D

The amplitude is the absolute value of A; |A|. It is the distance from the sinusoidal axis to a maximum (or minimum). If it is negative, the graph is reflected (flips) over the sinusoidal axis.

The Role of Parameter A ƒ(x) = AsinB(x - C) + D

B is not the period; it determines the period according to this relation: The Role of Parameter B

or

ƒ(x) = AsinB(x - C) + D

C is called the phase shift, or horizontal shift, of the graph.

The Role of Parameter C ƒ(x) = AsinB(x - C) + D

WATCH THE SIGN OF C

when C > 0 the graph shifts right

when C < 0 the graph shifts left

ƒ(x) = AsinB(x - C) + D

ƒ(x) = asin(bx - c) + dc = BC

In general form, the equation and graph of the basic sine function is:

ƒ(x) = AsinB(x - C) + D

In general form, the equation and graph of the basic cosine function is:

ƒ(x) = AcosB(x - C) + D

2π

2π

-2π

-2π

-π

-π π

πSince these graphs are so similar (they differ only by a "phase shift" of π/2 units) we will limit our study to the sine function.

The "starting point."

The "starting point."

Note that your calculator displays: ƒ(x) = asin(bx - c) + d

Which is equivalent to: ƒ(x) = AsinB(x - c/b) + D

A=1, B=1, C=0, D=0

A=1, B=1, C=0, D=0

How many revolutions (in radians and degrees) are illustrated in each graph? How many periods are illustrated in each graph?

Periods = Radians Rotated = Degrees Rotated =

Periods = Radians Rotated = Degrees Rotated =

Periods = Radians Rotated = Degrees Rotated =

HOMEWORK

Determine approximate values for the parameters 'a', 'b', 'c', and 'd' from the graphs, and then write the equations of each graph as a sinusoidal function in the form: y = a sin b(x + c) + d HOMEWORK

ƒ(x) = AsinB(x - C) + D

Determine approximate values for the parameters 'a', 'b', 'c', and 'd' from the graphs, and then write the equations of each graph as a sinusoidal function in the form: y = a sin b(x + c) + d HOMEWORK

ƒ(x) = AsinB(x - C) + D

State the amplitude, period, horizontal shift, and vertical shift for each of the following:

amplitude: period: horizontal shift:vertical shift:

amplitude: period: horizontal shift:vertical shift:

HOMEWORK

State the amplitude, period, horizontal shift, and vertical shift for each of the following:

amplitude: period: horizontal shift:vertical shift:

amplitude: period: horizontal shift:vertical shift:

HOMEWORK

Enter the values into your calculator, and use a sinusoidal regression to determine the equation. Round the values of the parameters to one decimal place.

x -1 -0.5 0 0.5 1 1.5 2 2.5 y 1 -2.6 -5.6 -5.4 -2 1.4 1.6 -1.4

HOMEWORK