A quadratic function is a polynomial function of degree 2. It has the general form f of x equals a x squared plus b x plus c, where a, b, and c are constants, and a cannot be zero. The graph of a quadratic function is a parabola, which has a characteristic U-shape or inverted U-shape.
A quadratic function has three main components. The coefficient a determines the shape and direction of the parabola. The coefficient b affects the horizontal position, and the constant c represents the y-intercept. The key requirement is that a cannot equal zero, otherwise it would become a linear function instead of quadratic.
Parabolas have several key features. The vertex is the highest or lowest point of the parabola. The axis of symmetry is a vertical line that passes through the vertex. When the coefficient a is positive, the parabola opens upward, and when a is negative, it opens downward. The y-intercept occurs at the point zero comma c.
Here are some common examples of quadratic functions. The basic parabola f of x equals x squared opens upward. When we have negative x squared, the parabola opens downward. Adding linear and constant terms shifts the parabola's position. Quadratic functions appear in many real-world applications, including projectile motion in physics, profit optimization in economics, and solving area and volume problems.
To summarize, a quadratic function has the general form f of x equals a x squared plus b x plus c, where a cannot equal zero. The graph is always a parabola with a vertex and axis of symmetry. Quadratic functions are fundamental in algebra and have countless real-world applications in physics, engineering, economics, and many other fields.