Processing math: 100%
WTMaths logo
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=4x2+4x-4.

Rewrite the equation y=4x2+4x-4 in completed square form:

y=(2x+1)2-5

The turning point is where (2x+1)=0 or x = -12

When x=-12, y=-5.

Answer: (-12,-5)

Example 2

By completing the square, determine the y value for the turning point for the function f(x)=x2+4x+7

Complete the square:

x2+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

See also Completing the Square