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On a parabola, the sole vertex lies on the axis of symmetry and in a quadratic of the form: a x 2 + b x + c {\displaystyle ax^{2}+bx+c\,\!} it can be found by completing the square or by differentiation . [ 2 ]
The vertex of a parabola is the place where it turns; hence, it is also called the turning point. If the quadratic function is in vertex form, the vertex is (h, k). Using the method of completing the square, one can turn the standard form = + + into
In contrast, the graph of the function f(x) + k = x 2 + k is a parabola shifted upward by k whose vertex is at (0, k), as shown in the center figure. Combining both horizontal and vertical shifts yields f(x − h) + k = (x − h) 2 + k is a parabola shifted to the right by h and upward by k whose vertex is at (h, k), as shown in the bottom figure.
If a < 0, the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The x-coordinate of the vertex will be located at =, and the y-coordinate of the vertex may be found by substituting this x-value into the function.
In the theory of quadratic forms, the parabola is the graph of the quadratic form x 2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x 2 + y 2 (or scalings), and the hyperbolic paraboloid is the graph of the indefinite quadratic form x 2 − y 2. Generalizations to more variables yield ...
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