Student worksheet – Teacher version
Quadratic modelling of the number of males registered in apprenticeship programs

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 minimum number of registrations of males in apprenticeship programs occurred.

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

   By inspection, h appears to be approximately 1995.

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 minimum number of registrations by males in apprenticeship programs.

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

   By inspection, k appears to be approximately 151 000.

7. What is the difference in the parabola between 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 = 1749.

13. What is your best value for h?

    Answers vary. In this example, h = 1995.32.

14. What is your best value for k?

    Answers vary. In this example, k = 150 975.

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

    y = 1749(x - 1995.32)² + 150 975

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 (1995.32, 150 975). This means that the lowest number of registrations of males in apprenticeship programs (150 975) occurred approximately in the year 1995.

18. Paste your graph here.