Applications of Sinusoidal Functions



Applications of Sinusoidal Functions

The equation of a sinusoidal function can be determined by examining the properties of the graph. Consider the following:

• The amplitude ‘a’ is half of the distance between the maximum and minimum: [pic]

It is the distance from equilibrium to the peak.

• The value for ‘k’ can be determined by observing the period (one cycle) from peak to peak: [pic]

• The value ‘c’ is the horizontal displacement to the line/axis of equilibrium; ‘c’ is positive if it is moved up and negative if it is moved down. We can also use the equation [pic]

• There are multiple options for the phase shift ‘d’. Look to see

how far a starting point of the sine/cosine curve on the

equilibrium line has been moved to the right. Hint:

1. For applications where the function starts from an extrema (maximum or minimum), use a cosine curve.

2. For applications where the function starts from a point in the middle (at equilibrium), use a sine curve.

Example 1

Determine an equation for each sinusoidal graph below:

a) b)

Example 2

The first Ferris wheel built in 1893 had a diameter of 80 m. The base of the wheel was 5 m above the ground. It took 20 minutes to do 2 full revolutions. If Milton boarded the bottom of the wheel at 9:07, how high from the ground would he be at 9:20?

[pic]

Example 3

The top of a building designed by Fractals Incorporated sways left (-10 m) then right (10 m) of its stable equilibrium position. One full swing from equilibrium to the left, to the right then back to equilibrium takes 4 seconds.

At t= 1s, the building is at equilibrium and about to swing left. What is the displacement at the top of the building at 3.5 seconds?

HMWK: pg398 # 1, 2, 4, 5, 11, 13

-----------------------

[pic]

[pic]

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download