© Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Completing the square

means writing the unknown terms of a quadratic in a square bracket Completing the square because Application To find the maximum or minimum value of this function. Example Think about… What is always true about the value of ( x + 3) 2 ? What will always be true for ( x + 3) 2 – 2? Minimum value ofis –2

Application To find the minimum point on the graph of the function. has a minimum valuewhen x = –3 Minimum point on the curve is (–3, –2) Think about… For what value of x is ( x + 3) 2 – 2 a minimum? Think about… What shape is the graph of y = x x + 7 ? Can completing the square help you sketch the graph of a quadratic?

0 y x 7 (–3, –2) line of symmetry x = –3 Minimum point is (–3, –2) When x = 0, y = 7 x = –3 is a line of symmetry So the intercept on the y axis is 7 Graph of y = x x + 7

0 y x 5 has a minimum valuewhen x = 1 Example Minimum point is (1, 3) To sketch the graph of Intercept on the y axisis 5 (1, 3) line of symmetry x = 1 Think about… What is the maximum or minimum value? Think about… What shape will the graph be? Where is its turning point?

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 Think about… What is the maximum or minimum value? Think about… What shape will the graph be? Where is its turning point? Note you can find the intercepts on the x axis by solving

Reflect on your work How does completing the square help you to sketch the graph of the function? Can you use completing the square to tell you whether the quadratic function has any real roots? Completing the square