1. Means writing the unknown terms of a quadratic in a square bracket Completing the square Example 1 This way of writing it is very useful when trying to sketch the curve and finding the vertex = (x + 3)(x + 3) –2 = x 2 + 3x + + 9 – 2 = x 2 + 6x + 7

2. Means writing the unknown terms of a quadratic in a square bracket Completing the square Example 1 We require x 2 + 6x + 7 Not x 2 + 6x + 9 So subtract 2 Halve the coefficient of x and square the bracket

3. Completing the square Example 2 We require x 2 - 4x + 9 Not x 2 - 4x + 4 So add 5 Halve the coefficient of x and square the bracket

4. Completing the square Example 3 We require x 2 - 4x Not x 2 - 4x + 4 So subtract 4 Halve the coefficient of x and square the bracket

5. Completing the square Example 4 We require x 2 - 4x Not x 2 - 4x + 4 So subtract 4 Halve the coefficient of x and square the bracket Change the signs in the bracket and change the sign outside

6. Sketching the graph Application To find the maximum or minimum value of this function. has a minimum value when x = – 3 Example Minimum value ofis y = – 2 Minimum point on the curveis (– 3, – 2) (– 3,– 2) X

7. Sketching the graph Example Minimum point on the curve is (– 3, – 2) The curve has been translated horizontally by -3 The curve has been translated vertically by -2 y = x 2 y = (x + 3) 2 -2 -2 -3 -4 -5 1 2 3 4 5 -2-3-4123450 X-> ^ | Y -2 -3 -4 -5 1 2 3 4 5 -2-3-4123450 X-> ^ | Y

8. Sketching the graph Example Minimum point on the curve is (– 3, – 2) The horizontal translation is the opposite sign to the term INSIDE the bracket i.e -3

9. Sketching the graph Example Minimum point on the curve is (– 3, – 2) The vertical translation is the same sign as the term OUTSIDE the bracket i.e -2

10. Sketching the graph Example Maximum point on the curve is (2, 4) The vertical translation is the same sign as the term OUTSIDE the bracket i.e +4 The horizontal translation is the opposite sign to the term INSIDE the bracket i.e +2 The negative sign outside the bracket has meant the curve flips vertically -2 -3 1 2 3 4 5 123450

11. Example Take the coefficient of x 2 outside the bracket containing the x 2 and x terms Now consider the term inside the bracket x 2 – 2x Halve the coefficient of x and square the bracket We require x 2 - 2x Not x 2 - 2x + 1 So subtract 1 Multiply by the 2

12. has a minimum value when x = 1 Minimum point is (1, 3) To sketch the graph of Intercept on the y axisis 5 (1, 3) line of symmetry x = 1 y = x 2 y = 2(x-1) 2 + 3

13. has a maximum valuewhen x = – 2 Example Maximum point is ( – 2, 7) 0 y x 3 To sketch the graph of Intercept on the y axisis 3 (–2, 7) x = –2 (Note you can find the intercepts on the x axis by solving )