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 – 2 = x 2 + 6x + 7

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

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

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

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

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

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) X-> ^ | Y X-> ^ | Y

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

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

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

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

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

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 )