Foundations 12 Sections 8.3

The Graphs and Equations of Sinusoidal Functions

Foundations 12 Sections 8.3

Investigating Sinusoidal Functions

Students in a graduating class went on an exchange trip to China. While there, they rode the Star of Nanchang, one of the tallest Ferris wheels in the world. The sinusoidal function that represents the ride is shown in the graph below.

Foundations 12 Sections 8.3

1. Determine how far the lowest part of the Ferris wheel is off the ground.

2. Determine the height of the Ferris wheel.

3. Determine the range of the graph. What does it represent?

4. Determine the amplitude of the graph. What does it represent?

5. Determine the equation of the sinusoidal axis. What does the sinusoidal axis represent?

6. Determine the period of the graph. What does it represent?

7. How long does it take to get to the top of the Ferris wheel from the bottom?

Foundations 12 Sections 8.3

Example 1: Describing the Graph of a Sinusoidal Function In Degree Measure

The graph of a sinusoidal function is illustrated above. Describe this graph by determining its range, the equation of its sinusoidal axis, its amplitude and its period.

Example 2: Describing the Graph of a Sinusoidal Function in Radian Measure

Determine the range, amplitude, period and equation of the sinusoidal axis for the sinusoidal function illustrated to the left.

Example 3: Connecting a Sinusoidal Function to Oscillating Motion

Foundations 12 Sections 8.3

For a physics project, Karley and Fiona had to graph and analyze an example of simple harmonic motion. Karley swung on the swing, and Fiona used a motion detector to measure Karley’s height above the ground over time, as she swung back and forth. The girls then graphed their data as shown. At the end of each cycle, the swing returned to its initial position, which resulted in a sinusoidal graph.

a. Determine the range of the graph? What does it represent with respect to the swing?

b. Determine the equation of the sinusoidal axis.

c. Determine the amplitude of the graph. What does it represent with respect to the swing?

d. Determine the period of the graph. What does it represent with respect to the swing?

Example 4: Comparing Two Sinusoidal Functions

Harrison and Courtney own a car and a pickup truck. The noticed the odometers of the two vehicles gave different values for the same distance. As part of their investigation into the cause, they put a chalk mark on the outer edge of a tire on each vehicle. The following graphs show the height of the tires as they rotated while the vehicles were driven at the same slow, constant speed. What can you determine about the characteristics of the tires from the graphs?

Foundations 12 Sections 8.3

a. What is the minimum value on each graph? What does it represent?

b. Determine the maximum value on each graph. What do they represent in this context?

c. Determine the equation of the sinusoidal axis. What do they represent in this context?

Foundations 12 Sections 8.3

d. Determine the period of each graph. What do the periods indicate in this context?