Completing the Square

1. Completing the Square

2. Learning Goal • Transform the standard form of a quadratic relation into VERTEX FORM by completing the square • Given a quadratic relation in VERTEX FORM, state the MAXIMUM or MINIMUM VALUE and when it occurs(the vertex) and sketch the graph

3. Investigation on p.124 • 1. c)

4. #2

5. Final Answer:

6. Algebraic Method for #4

7. METHOD: ex. • 1. Take half of the x term and square it. • 2. Add and subtract that term. • 3. Factor the first three terms as a perfect square.

8. If the coefficient of the first term is not one, you have to COMMON FACTOR OUT THIS COEFFICIENT • Ex.

9. Application Problem: • The path of a ball is modelled by • Where x is the horizontal distance and y is the height, both in metres • Determine the maximum height of the ball and state at what horizontal distance it occurs.

11. FINAL ANSWER: • We know it is a maximum height because the leading coefficient is negative, and therefore the parabola opens down. • Therefore the maximum height is 4m and this occurs when the horizontal distance is 1m.

12. GRAPH:

13. Maximum and Minimum Points • For • the vertex is (h,k) • 1) If a > 0, then • A) the graph opens up • B) (h, k) is a minimum point • C) k is the minimum value and occurs at h

14. Maximum and Minimum Points • For • 2) If a < 0, then • A) the graph opens down • B) (h, k) is a maximum point • C) k is the maximum value and occurs at h

15. HOMEWORK: • p.270 • #1a,3acg,4bc,5,6a, • 7ace,8c,10d,12,15