Applied 40S May 25, 2009

Learning to Read the Wave

Riding the Perfect Wave by flickr user San Diego Shooter

Generate a sinusoidal equation and sketch the graph for the following set of data. Write the values of the parameters in your equation to two decimal places. Homework

Source: Winnipeg weather statistics

Month J F M A M J J A S O N Dmm 19 15 23 36 60 84 72 75 51 30 21 19

Amount of Precipitation in Winnipeg (in mm)

Generate a sinusoidal equation and sketch the graph for the following set of data. Write the values of the parameters in your equation to two decimal places. Homework

Year 0 1 2 3 4 5 6 7 8Population 200 188 160 132 120 133 161 187 201

The data below show the population of a species of marmot in a given area over a 9-year period on June 1st of each year.

Properties and Transformations of the sine function ...

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

ƒ(x) = AsinB(x - 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.

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

y = 2sin(x)

y = -3sin(x)

y = 1 sin(x) 2

Properties and Transformations of the sine function ...

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

ƒ(x) = AsinB(x - C) + D

y = 2sin(x) + 3

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

or

y = sin(3x)

y = sin(2x)

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

The Role of Parameter C

y = sin(x - π ) 4

y = sin(x + π ) 4

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

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=02π-2π -π π

The "starting point."

ƒ(x) = AsinB(x - C) + D

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 Remember

"DABC!"

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