Quadratic Function
f(x) = a(x – h)^2 + kDomain: ]-∞; ∞[Range: [k; ∞[
A quadratic function is a function that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a is not equal to 0. This function appears as a parabola when graphed and has various properties.
The coefficient a determines the shape of the parabola. If a is positive, the parabola opens upwards, and if it is negative, the parabola opens downwards. The vertex of the parabola is located at the point (-b/2a, f(-b/2a)) and is the lowest or highest point depending on the direction of the parabola.
The x-intercepts (also called roots or zeros) of the quadratic function can be found by setting f(x) = 0 and solving the resulting quadratic equation. If the discriminant b^2 – 4ac is positive, then the function has two real roots, if it is zero, the function has a single real root, and if it is negative, the function has two complex (non-real) roots.
Applications of quadratic functions include modeling the trajectory of projectiles, finding the maximum or minimum value of a function, and solving optimization problems.
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