Quadratic Function In Standard Form

The standard form of a quadratic function is given by: f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0.

The vertex (h, k) of the quadratic function f(x) = ax^2 + bx + c can be found using the formula: h = -b/(2a) and k = f(h).

The vertex represents the maximum or minimum point of the quadratic function, depending on the direction of the parabola (upward or downward).

A quadratic function opens upwards if the coefficient 'a' is positive (a > 0) and opens downwards if 'a' is negative (a < 0).

The y-intercept of a quadratic function f(x) = ax^2 + bx + c is the point where the graph intersects the y-axis, found by evaluating f(0) = c.

The x-intercepts can be found by solving the equation ax^2 + bx + c = 0 using factoring, completing the square, or the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).

The quadratic formula is used to find the roots of a quadratic equation ax^2 + bx + c = 0 and is given by: x = (-b ± √(b^2 - 4ac)) / (2a).

The discriminant, given by D = b^2 - 4ac, indicates the nature of the roots: D > 0 means two distinct real roots, D = 0 means one real root, and D < 0 means no real roots.

The maximum or minimum value of a quadratic function occurs at the vertex and can be found by evaluating f(h) where h = -b/(2a).

To convert from standard form f(x) = ax^2 + bx + c to vertex form f(x) = a(x - h)^2 + k, complete the square.