Turning Points from Completing the Square

# Turning Points from Completing the Square

GCSE(H),

A turning point can be found by re-writting the equation into completed square form (see Completing the Square).

When the function has been re-written in the form y = r(x + s)^2 + t, the minimum value is achieved when x = -s, and the value of y will be equal to t.

The coordinate of the turning point is (-s, t).

## Examples

1. What is the coordinate of the turning point for the equation y = 4x^2 + 4x - 4?

Answer: (-frac(1)(2) -5)

Rewrite the equation y = 4x^2 + 4x - 4 in completed square form:

y = (2x + 1)^2 - 5

The turning point is where (2x + 1) = 0 or x = frac(-1)(2)

When this is true, y = -5.

2. What is the coordinate of the turning point for the equation y = x^2 + 4x + 7?