Turning Points from Completing the Square

## Turning Points from Completing the Square

A turning point can be found by re-writting the equation into completed square form.

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).

## Example 1

By completing the square, determine the coordinate of the turning point for the equation y = 4x^2 + 4x - 4.

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 x=-frac(1)(2), y = -5.

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

## Example 2

By completing the square, determine the y value for the turning point for the function f(x) = x^2 + 4x + 7

Complete the square:

x^2 + 4x + 7 = (x+2)^2 + 3

When x = -2, the bracket evaluates to zero, leaving a residual value of 3. This is the y value of the turning point.

Answer: y=3