Student worksheet – Teacher version
Sinusoidal modelling of Canada's youth cohorts

1. What do the following variables represent? Provide both the term and a full explanation of the term.
   • a represents the amplitude of the sine curve, which is the maximum distance from the centre line.
   • k represents the period of the curve, which is the length of time to complete one wavelength of the curve.
   • d represents the phase (horizontal shift) of the curve, which is the number of units the sine curve has moved from its starting position to the left or right.
   • c represents the displacement (vertical shift) of the curve, which is the number of units the sine curve has moved from its initial position up or down.
2. Provide your best values for the parameters below and an explanation of what they mean for this dataset.
   • a
     Answers vary – 250,000 used in this example. This means that, from the middle value (c), the curve varies by 250,000 both upwards and downwards. In other words, the number of 20 to 24 year-olds varies from the middle value by 250,000.
   • k
     Answers vary – 27.3 used in this example. This means that the period of the graph is 27.3 years. In other words, it takes 27.3 years for the cycle to repeat itself. This corresponds to the length of one generation.
   • d
     Answers vary – 5.00 used in this example. This means that this graph is shifted from a basic sine curve by 5.00 units (years) to the right.
   • c
     Answers vary – 2,250,000 used in this example. This means that the graph is shifted from a basic sine curve by 2,250,000 units up. In other words, the centre line is now at y = 2,250,000 instead of being the x-axis (y = 0).
3. What is your equation for this graph, in the form ?
   Answers vary. In this example: .
4. By inspection of your graph, what is the approximate range for this dataset (to the nearest hundred thousand)?

   The approximate range is 2,000,000 y 2,500,000, which constitutes a range of 500,000.

5. If you know the amplitude for a sine curve, how can you calculate the range?
   Calculate the range here, based on the amplitude from your graph.
   

   Range = 2 x Amplitude
   Range = 2(250,000)
   Range = 500,000

6. What happens to the graph when:
   1. the value of a is increased?
      The amplitude of the graph increases, which in turn increases the range.
   2. the value of k is decreased?
      The graph is stretched horizontally, which increases the wavelength/period.
   3. the value of d is negative? Is this reasonable for this dataset? Why/why not?
      The graph has a horizontal phase shift to the left. This is reasonable because it is the shift from the basic sine curve, so it can be to the left or right and still be valid.
   4. the value of c is negative? Is this reasonable for this dataset? Why/why not?
      The centre line shifts vertically below the x-axis. This is not reasonable for this dataset because it would indicate a negative number of 20 to 24 year-olds, which is impossible.
7. How might these data be useful to administrators at a university or college?
   

   These data might be useful to administrators at a university or college to predict the number of students in the future. When there are more 20 to 24 year-olds, there will be more students in college or university. Thus, universities and colleges would need to be prepared for a greater number of students. For example, more spaces in residences will be needed and more professors may be needed in order to offer more sections of classes.

8. Paste your sine graph here.

   Fathom graph
   (In some provinces, Fathom is licensed by the Ministry of Education and used in schools. Providing the Fathom format is in no way an endorsement or recommendation of the Fathom software by Statistics Canada.)

   

   E-STAT graph
   To generate the graph below, select Line graph with symbols in step 9 of E-STAT instructions.

   

   Source: Statistics Canada. Table 051-0001 - Estimates of population, by age group and sex for July 1, Canada, provinces and territories, annual (persons unless otherwise noted) (graph), CANSIM (database), Using E-STAT (distributor).
   http://estat.statcan.gc.ca/cgi-win/cnsmcgi.exe?Lang=E&ESTATFile=EStat\English\CII_1_E.htm&RootDir=ESTAT/
   (accessed: April 30, 2008)