The following situation describes a new kind of function:
The ocean has high tides and low tides. The tide comes in for six hours (ending at "high tide") and then goes out for six hours (ending at "low tide"). This is repeated twice in a day. An approximate rule to describe the motion of the tide is this:
From low to high tide:
In the first hour, 1/12 of the tide comes in.
In the second hour, 2/12 of the tide comes in.
In the third hour, 3/12 of the tide comes in.
In the fourth hour, 3/12 of the tide comes in.
In the fifth hour, 2/12 of the tide comes in.
In the sixth hour, 1/12 of the tide comes in.
From high to low tide:
In the first hour, 1/12 of the tide goes out.
In the second hour, 2/12 of the tide goes out.
In the third hour, 3/12 of the tide goes out.
In the fourth hour, 3/12 of the tide goes out.
In the fifth hour, 2/12 of the tide goes out.
In the sixth hour, 1/12 of the tide goes out.
Let's say the height at low tide is zero and at high tide is 24 feet.
Remember that full tide is 24 feet while calculating the tide for each hour. Close Tip
Make a graph of the hours after low tide vs. the height of the tide. Connect the points with a smooth curve.
Extend your graph to show a full day (24 hours) of tides.
Think carefully about what would happen during the second 12 hours. Close Tip
Describe how this graph is different from graphs of linear, exponential, and quadratic functions you've seen.
Functions like the one you just graphed here are called cyclic functions, also known as repeating or periodic functions. These functions are characterized by outputs that repeat in a cycle. Cyclic functions are important in astronomy (they're used to describe the motion of the planets), engineering, and many other fields.
Two important characteristics of cyclic functions are amplitude and period. The amplitude measures the height of the graph. It's defined this way:
amplitude = (highest point - lowest point) / 2
The period of a cyclic function is how long it takes to complete a cycle.
What is the period of the graph in Problem A3?
Video Segment In this video segment, the cyclic function presented in Part A is graphed. The class then discusses the definitions of amplitude and period. Watch this segment after you've worked on Problems A1-A5. If you get stuck on the problems, you can watch the video segment to help you.
Think of some other cyclic functions you might encounter in everyday life. What are their periods?
You can find this segment on the session video, approximately 4 minutes and 3 seconds after the Annenberg Media logo.
Video Segment In this video segment, taken from the "real world" example at the end of the Session 8 video, Rick Garnen of the Massachusetts Maritime Academy talks about the causes and effects of tides on currents in Cape Cod Bay and how this relates to the mathematics of cyclic functions.
You can find this segment on the session video, approximately 23 minutes and 8 seconds after the Annenberg Media logo.
Tide problem adapted from Trigonometry, by I. M. Gelfand and Mark Saul (Boston: Birkhauser Publishing Ltd., 2001). ISBN:0-8176-3914-4.