Find Turning Point by Completing the Square
The turning point of a parabola is its highest or lowest point. Completing the square is the most reliable algebraic method to find these coordinates.
Find Turning Point
What is a Turning Point?
The turning point, or vertex, is where the quadratic function changes direction from decreasing to increasing (a minimum) or increasing to decreasing (a maximum). If the leading coefficient 'a' is positive, it's a minimum. If 'a' is negative, it's a maximum.
Reading the Coordinates
Once you have completed the square to get y = a(x - h)² + k, the turning point is simply the coordinate pair (h, k). Note that the sign of 'h' is flipped from what appears inside the parentheses.
Alternative Methods
While you can also find the x-coordinate of the turning point using the formula x = -b/(2a) or by taking the first derivative and setting it to 0, completing the square provides both the x and y coordinates simultaneously in one algebraic transformation.
Complete the Square Calculators
Specialized tools for every completing the square scenario — pick the one that matches your problem.