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`?

Answer: (-2, 3)

(x + 2)(x + 2) = x2 + 4x + 4

Therefore (x + 2)2 + 3 = x2 + 4x + 7

Make one of the brackets equal to zero; x = -2. y is equal to the remainder.

Turning point is at (-2, 3).