The Quadratic Parent Function: Coefficients, Vertex, And Applications

Quadratic Parent Function

Domain: [0, ∞) Range: [0, ∞)

The quadratic parent function is a mathematical equation that graphs as a parabola in the coordinate plane. The general format of the quadratic parent function is f(x) = ax² + bx + c, where a, b, and c are constants that determine the shape, position, and orientation of the parabola.

The coefficient a represents the vertical stretch or compression of the parabola. If a > 1, then the parabola is vertically stretched, while if 0 < a < 1, it is vertically compressed. If a is negative, the parabola is reflected across the x-axis. The coefficient b represents the horizontal shift and is responsible for moving the axis of symmetry from the y-axis to a different location. A positive value of b shifts to the left, and a negative value shifts to the right. The coefficient c represents the vertical shift of the vertex. If c > 0, the vertex is shifted up, and if it is negative, the vertex is shifted down.

The vertex of the quadratic parent function is (-b/2a, c) and the axis of symmetry is x = -b/2a. The vertex is the point on the parabola with the minimum or maximum y-value, depending on whether the parabola opens up or down.

The quadratic parent function can be used to model many real-world situations, such as projectile motion or the shape of a satellite dish. It is also the foundation for solving quadratic equations and for the study of algebraic functions.

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