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Modelling Periodic Behaviour
The largest Ferris Wheel opened in Chaoyang Park, Beijing just in time for the 2008 Olympics. The 682­foot­high wheel, which has its centre 346 feet above the ground, will give up to 5,760 passengers per hour a fantastic view of the city and surrounding
area. Each of the wheel's 48 capsules holds 40 people. Suppose that you and a group of friends are
riding the Ferris wheel. The ride then begins with you at point A. The
Ferris wheel turns counterclockwise at a constant speed. The wheel
takes one minute to complete one revolution.
1. State the maximum and minimum height of the Ferris Wheel.
687 feet
A
5 feet
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2. Identify the coordinates, in terms of time (seconds) and height
(feet) of the points B, C, and D. Think in terms of a ball of string
unravelling, creating a wave.
C
D
682 ft
A
B
(60, 5)
3. Complete the following table by giving your height relative to the x­axis after
the given rotations.
Rotation of Wheel
(seconds)
0
15
30
45
60
Height, relative to x­axis (feet)
2
4. Plot the points on a grid. Sketch a curve of best fit to show the relationship
between your height, h, and the time, t.
height
0
15
30
45
60
time
3
Key Ideas
• A function is periodic if it has a pattern of y­values that repeat
at regular intervals.
• One complete pattern is called a cycle.
• The horizontal length of one cycle is called the period of the function.
• The horizontal line that is halfway between the maximum and
minimum values of a periodic curve is called the axis of the curve.
• The equation of the axis of the curve is
y = maximum value + minimum value
2
• The magnitude of the vertical distance from the axis of the
curve to either the maximum or minimum value is called the
amplitude of the function.
• The amplitude, a, is calculated as
a = maximum value ­ minimum value
2
Note: A cycle may begin at any point of the graph.
One cycle = One period
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Example 1: Determine if the function is periodic. If so,
state the period, the maximum value, the minimum value,
the amplitude and the equation of the axis of the curve.
a)
b)
c)
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Example 2: Determine whether each graph is periodic or not.
a) tip of vibrating meter stick
b) movement of a piston in a combustion engine
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Example 3:
The automatic dishwasher in a school cafeteria runs constantly through lunch.
The graph shows the amount of water used as a function of time.
a) Explain why the operation of the dishwasher is an example of a periodic
function.
The graph of the dishwasher repeats and each cycle is the same.
b) What is the length of the period? What does one complete cycle mean in
the context of the question?
The period is the length of one complete cycle. Period = 15
One complete cycle could mean an 8 min wash, followed by a 5 min rinse,
followed by 2 min to unload and load dishes.
c) Extend the graph for one more complete cycle.
d) How much water is used if the dishwasher runs through eight
complete cycles?
The dishwasher uses 15 L to wash and another 15 L to rinse.
Then eight cycles uses 8(15 + 15) = 240 L
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Example 4: The Bay of Fundy, which is between New Brunswick and Nova
Scotia, has the highest tides in the world. There can be no water on the beach
at low tide, while at high tide the water covers the beach.
a) Why can you use a periodic function to model the tides?
Tides resemble periodic functions because they repeat over a fixed
interval of time.
b) What is the change in the depth of water from low tide to high tide?
The water level at low tide is zero. The water level at high tide is
10 m. The change is 10 m.
c) Determine the equation of the axis of the curve.
d) What is the amplitude of the curve?
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Example 5:
The function shown is periodic. Answer
questions 1 ­ 3 for this function.
1. Determine the period of the function.
2. Find:
a) f(4)
b) f(5)
c) f(8)
d) f(13)
3. Determine the amplitude of the function.
Example 6:
Suppose you were not given a graph, but told that a periodic function f has a period of 9. If f(12) = 8 and f(15) = ­1 determine f(48) and f(­3).
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Homework
Text pg 352 #1­10
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