applied math 40s may 23, 2008

The "periodic" moments of our lives ...

orTransformations of the sine function

Sunshine Coast Panoramic by flickr user El Fotopakismo

The sine curve (graph) ... HOMEWORK

HOMEWORK

HOMEWORK

HOMEWORK

HOMEWORK

http://www.poodwaddle.com/worldclock.swf

Let's look at the weather ...

Month J F M A M J J A S O N DMean -17 -14 -6 4 12 17 20 18 12 6 -4 -14

Winnipeg Weather Data as of May 15, 2007 for the last year

Temperature

Source: Winnipeg weather statistics

Source: Winnipeg weather statistics

Month J F M A M J J A S O N DMean 120 140 178 232 277 291 322 286 189 150 95 99

Hours of Sunshine

swivel your data

Properties and Transformations of the sine function ...

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

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

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.

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

or

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

The Role of Parameter C

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