Applications of Periodic Functions

orBugs On Wheels

Suicidal Shield Bug by flickr user ChinchillaVilla

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

How many periods are illustrated in each graph?

Periods = 1.5Radians Rotated = 3πDegrees Rotated = 540°

Periods = 3.25Radians Rotated = 6.5πDegrees Rotated = 1170°

Periods = 0.75Radians Rotated = 1.5πDegrees Rotated = 270°

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

y = 6sin(πx) + 73

y = 4sin(πx) - 4

y = 2sin(π(x - 1))3

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

amplitude: 2

period: 2π 3horizontal shift: +4

vertical shift: +1

amplitude: 4

period: 2

horizontal shift: 0

vertical shift: 0

HOMEWORK

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

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

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

Since these graphs are so similar (they

differ only by a "phase shift" of π units) 2 we will limit our study to the sine function.

2π-2π

-π π

The "starting point."

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

ƒ(x) = AcosB(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:

Jud was working with sinusoidal data, but lost all of it except for two points. A maximum point was (3, 13) and a minimum point next to it was (7, 1). Write a sinusoidal equation that matches Jud's data.

ƒ(x) = AsinB(x - C) + D

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 2.5y 1 -2.6 -5.6 -5.4 -2 1.4 1.6 -1.4

HOMEWORK

At a sea port, the depth of the water, h meters, at time t hours after midnight, during a certain day is given by this formula:

(a) State the: (i) period (ii) amplitude (iii) phase shift.

(b) What is the maximum depth of the water? When does it occur?

(c) Determine the depth of the water at 5:00 am and at 12:00 noon.

(d) Determine one time when the water is 2.25 meters deep.

HOMEWORK

The Bug on the Water Wheel A water wheel with a 7.0 ft radius has 1.0 ft. submerged in the water as shown, and rotates counterclockwise at 6.0 revolutions per minute. A bug is sitting on the wheel at point B. You start your stopwatch, and two seconds later the bug at point B is at its greatest height above the water. You are to model the distance 'h' of the bug from the surface of the water in terms of the number of seconds 't' the stopwatch reads.

(a) Sketch the graph.(b) Write the algebraic equation of the sinusoid.(c) How far is the bug above the water when t = 5.5 seconds?

HOMEWORK

Apogee and Perigee A spacecraft is in an elliptical orbit around the earth as shown in the diagram. At its perigee the spacecraft is at its lowest point above the earth's surface, and at its apogee it is at its highest point above the earth.

The equation: describes the

spacecraft's distance from the earth over time, where 'x' represents the time in minutes, and 'y' the distance in kilometres. Write answers for the following questions:

(a) How long does it take for the spacecraft to fly around the earth?

(b) What are the distances at apogee and perigee?

HOMEWORK

The Water Heater and the Hot Tub An electric heater cuts in and out on a cyclical basis as it heats the water in a hot tub. The water temperature 'C' in degrees Celsius varies sinusoidally with time 't'. The thermostat is set to turn the heater on when the temperature drops to 36°C and off when it reaches 42°C. In order to measure the period, a timer was started at 0 when the heater cut in. The heater turned off 30 minutes later and turned on again after another 30 minutes.

(c) Use your equation to determine the temperature 10 minutes after the heater turns on.

(b) Write the equation expressing temperature, C, in terms of time, t.

(a) Sketch the graph of the function for 0 ≤ t ≤ 60.

HOMEWORK