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)`.
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`.
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.
See also Completing the Square