1. Lesson 4

1.5. Explore

Mathematics 30-2 Module 6

Module 6: Sinusoidal Functions

 

Explore

 

In Try This 1, you saw that it is possible to sketch a curve of best fit for sinusoidal data. You can then use the curve of best fit to predict the parameters of a sinusoidal equation. In the daylight example, both interpolation and extrapolation were reasonable because the data followed a sinusoidal pattern and you expect each year to look similar. Care needs to be taken when determining and interpreting a line or curve of best fit. You will explore this line or curve of best fit matter further in Try This 2.

 

Try This 2

 

A tide graph predicts the height of water at a particular location at particular times. Consider the following tide graph for Vancouver from October 7 to 11, 2011.

 

This graph shows the hourly depth of water over time near Vancouver. Water depth oscillates between approximately 1.5 metres and 4.5 metres.

SOURCE: Tide, Currents, and Water Levels, (Fisheries and Oceans Canada, 2011), <http://www.tides.gc.ca/>; (09/02/2012).

  1. Explain why a sinusoidal model is reasonable for this data.
  2. Use Tides Model 1 to determine an equation that matches the data. Adjust the parameters a, b, c, and d until the graph overlaps the points as closely as possible. Record your equation.

     
    This is a play button for Tides Model 1.
  3. Three points on the original scatter plot are (10, 1.9), (50, 3.0), and (100, 3.3). Calculate the y-value, to the nearest tenth, for the times of 10, 50, and 100 using your equation from question 2. When you’re making the calculations, be sure your calculator is in radian mode. Use a chart similar to the following one to organize the information. How close are the values predicted using an equation to the actual points?

    x-value

    y-value Using Equation Found in Question 2

    Actual y-value

    10

     

    1.9

    50

     

    3.0

    100

     

    3.3
  4. At midnight on January 6, 2012 (2184 h after midnight on October 7), the tide height was 1.9 m. How close does your equation predict this value?
  5. Consider the tide graph from the same location from November 16 to 20, 2011. What characteristics of this graph are similar to a sine graph? What characteristics are different?

    This graph shows the hourly depth of water over time near Vancouver. The time oscillates between approximately 1.5 m and 4.5 m.
    SOURCE: Tide, Currents, and Water Levels, (Fisheries and Oceans Canada, 2011), <http://www.tides.gc.ca/>; (09/02/2012).
  6. Use Tides Model 2 to try to model the data using an equation of the form

     
    This is a play button for Tides Model 2.
  7. Is it reasonable to use an equation of the form to model this data? Explain.

course folder Save your answers in your course folder.

 

Did You Know?

This photo shows the Moon rising over the surf.

iStockphoto/Thinkstock

The gravitational pull of the sun and moon are the main causes of coastal tides (water levels regularly changing). When the sun and moon are on the same side of Earth, or are opposite each other, tides tend to be larger and follow a pattern like the tides seen in Vancouver from October 7 to 11, 2011.

 

When the sun and moon are not on the same side of Earth, or are not across from one another, the tide can be thought of as two overlapping patterns like that seen in Vancouver from November 16 to 20, 2011.

Share 2

 

With a partner or in a group, discuss the following questions based on what you learned in Try This 2.

  1. How can you decide when a sinusoidal curve should be used to model data?
  2. Why are calculated values in question 3 of Try This 2 not the actual values that were given in the chart?
  3. What advantage is there in using an equation to represent data?
  4. Is it reasonable to use interpolation to predict values using your first graph? What about extrapolation? Explain.

course folder If required, save a record of your discussion in your course folder.