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)`.

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) = *x*^{2} + 4*x* + 4

Therefore (*x* + 2)^{2} + 3 = *x*^{2} + 4*x* + 7

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

Turning point is at (-2, 3).

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