Student worksheet – Teacher version
Quadratic modelling of Canada's Baby Boom

1. In general, what does h represent?

   h represents the x value at which the optimum (maximum or minimum) y value occurs. In other words, h is the x co-ordinate of the vertex.

2. Specific to this dataset, what does h represent?

   For this dataset, h represents the year in which the maximum number of births occurred.

3. By inspection of your scatter graph, what is a reasonable approximation of h?

   By inspection, h appears to be approximately 1960.

4. In general, what does k represent?

   k represents the optimum (maximum or minimum) y value. In other words, k is the y co-ordinate of the vertex.

5. Specific to this dataset, what does k represent?

   For this dataset, k represents the maximum number of births.

6. By inspection of your scatter graph, what is a reasonable approximation of k?

   By inspection, k appears to be approximately 480,000.

7. What is the difference in the parabola for a quadratic equation with a positive a value and a quadratic equation with a negative a value?

   If the a value is positive, the parabola opens up. If the a value is negative, the parabola opens down.

8. What happens if a is zero? Discuss in terms of both the equation and the resulting graph.

   If a is zero, the equation becomes y = k. This means that the resulting graph is a horizontal line at y = k.

9. If a quadratic equation has a positive a value, will the vertex of the parabola be a maximum or a minimum? Why?

   If the quadratic equation has a positive a value, the vertex of the parabola will be a minimum since the graph opens up.

10. If a quadratic equation has a negative a value, will the vertex of the parabola be a maximum or a minimum? Why?

    If a quadratic equation has a negative a value, the vertex of the parabola will be a maximum since the graph opens down.

11. How does changing the value of a affect the shape of a parabola?

    For positive a values, as a increases, the graph is stretched vertically (becomes narrower). When a is zero, the graph is a horizontal line (y = k). For negative a values, the graph is stretched vertically (becomes narrower) as the value of a decreases (becomes more negative). Essentially, as the absolute value of a increases, the graph becomes narrower.

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12. What is your best value for a?

    Answers vary. In this example, a = -1,545.

13. What is your best value for h?

    Answers vary. In this example, h = 1959.

14. What is your best value for k?

    Answers vary. In this example, k = 479,600.

15. Write your equation here in the form y = a(x – h)² + k.

    y = -1,545(x – 1959)² + 479,600

16. How are h and k related?

    h and k are the x and y values, respectively, of the vertex. That is, the vertex is the point (h, k).

17. What is the vertex of your graph? What does this mean in terms of this dataset?

    Answers vary. In this example, the vertex of the graph is (1959, 479,600). This means that the highest number of births (479,600) occurred in the year 1959.

18. Paste your graph here.

    Fathom graph
    (Fathom is licensed by the Ministry of Education and used by schools in some provinces. 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 8 of E-STAT instructions.
    
    
    Source: Statistics Canada. Table 053-0001 - Vital statistics, births, deaths and marriages, computed annual total (number) (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 10, 2008)