Understanding Quadratic Functions | Properties, Features, and Applications

Quadratic Function

A quadratic function is a type of polynomial function of degree 2

A quadratic function is a type of polynomial function of degree 2. It is defined by an equation of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a curve called a parabola.

Three key features of a quadratic function are:
1. Vertex: The vertex is the highest or lowest point on the graph of a quadratic function. If a > 0, the vertex is the lowest point, and if a < 0, the vertex is the highest point. The x-coordinate of the vertex can be found using the formula x = -b/(2a), and the y-coordinate can be found by substituting the x-coordinate into the function. 2. Axis of symmetry: The axis of symmetry is a vertical line that passes through the vertex. It divides the parabola into two symmetrical halves. The equation of the axis of symmetry is x = -b/(2a). 3. Roots or zeros: The roots or zeros of a quadratic function are the x-values where the graph intersects the x-axis. They represent the solutions to the quadratic equation ax^2 + bx + c = 0. The roots can be found using the quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a). The leading coefficient "a" in a quadratic function determines the direction of the parabola. If a > 0, the parabola opens upwards, and if a < 0, the parabola opens downwards. The coefficient "b" affects the position of the vertex and the symmetry of the parabola. Quadratic functions have various applications in real-life problems, such as determining the path of projectiles, modeling profit or revenue functions, optimizing shapes, and solving engineering problems.

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