Begin by sketching a graph of the y coordinate of a point moving around a circle at 3 rad/s vs. clock time t.

• How far is it between the peaks of the graph?
• Explain why, if we assume that the center of the circle lies on the x axis, the graph should be symmetric with respect to the x axis.
• Assuming that the center of the circle lies on the x axis, and that the circle has radius 5, then what is the maximum and what is the minimum y coordinate of a point on the graph?
• Sketch the corresponding graph.
• Explain why, if we assume that the center of the circle lies on the line y = 12, the graph should be symmetric about the line y = 12.
• Assuming that the center of the circle lies on the line y = 12 and that the circle has radius 5, the what is the maximum and what is the minimum coordinate of a point on the graph?

Where would you position the circle, and what would be its radius, if you wished to model a quantity y which varies cyclically from a value of 6 to a value of 14?

• Given the values between which a cyclical quantity varies, how you determine where to position the circle that models the quantity, and how the determine the radius of the circle?

At a certain latitude near here the length of a day varies from approximately 9 hours to approximately 15 hours over a period of 1 year, or 12 months.

• How could you position a circle so as to obtain a graph to model y = length of a day in hours vs. t = clock time in months?
• What angular velocity should you give the point moving around the circle?
• Sketch and label the corresponding graph.

The average daily mean temperature in this vicinity varies from an approximate maximum of 75 degrees Fahrenheit to an approximate minimum of 35 degrees Fahrenheit over a 52 week period.

• How could you position a circle so as to obtain a graph to model y = average daily mean temperature vs. t = clock time in weeks?
• What angular velocity should you give the point moving around the circle?
• Sketch and label the corresponding graph.

As the tide rolls in and out the water level varies from 12 feet below a certain walkway to 2 feet above, over a 10-hour period.

• How could you position a circle so as to obtain a graph to model y = water level vs. t = clock time in hours?
• What angular velocity should you give the point moving around the circle?
• Sketch and label the corresponding graph.

As the waves roll into the shore of the ocean at a certain place, a buoy bobs up and down from a point 40 feet above the ocean floor to a point 36 feet above the ocean floor, completing a full up and down cycle 5 times every minute.

• How could you position a circle so as obtain a graph to model y = buoy height above ocean floor vs. t = clock time in seconds?
• What angular velocity should you give the point moving around the circle?
• Sketch and label the corresponding graph.