Quadratic Functions: Definition, Graph, Vertex, Roots And Applications.

Quadratic Function

The graph is in the shape of a parabola.

A quadratic function is a type of nonlinear 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 zero.

The graph of a quadratic function is a parabola, which is a U-shaped curve. The direction of the opening of the parabola depends on the sign of the coefficient a. If a is positive, the parabola opens upward, and if a is negative, the parabola opens downward.

The vertex of the parabola is the minimum or maximum point of the function, depending on whether it opens upward or downward. The x-coordinate of the vertex is given by -b/2a, and the y-coordinate is obtained by plugging this value of x into the equation.

The roots or zeros of a quadratic function are the x-values where the graph intersects the x-axis. These can be found by solving the quadratic equation ax^2 + bx + c = 0 using the quadratic formula x = (-b ± √(b^2-4ac))/2a. The discriminant b^2-4ac determines the nature of the roots. If it is positive, there are two distinct real roots. If it is zero, there is one real root with a multiplicity of 2. If it is negative, there are two complex roots.

Applications of quadratic functions include modeling the motion of objects under constant acceleration (such as projectiles), finding the maximum or minimum value of a function subject to certain constraints, and solving optimization problems.

More Answers: