A parabola is a U-shaped curve that is commonly encountered in algebra and geometry. The equation of a parabola is derived from its geometric properties and the position of its vertex (the highest or lowest point) and focus. There are several forms of the parabola equation, each providing different insights into its properties.
Standard Form of a Parabola
The standard form of a parabola equation is y = ax^2 + bx + c, where:
- ‘a’ determines the width and direction of the parabola. If ‘a’ is positive, the parabola opens upwards; if ‘a’ is negative, it opens downwards.
- ‘b’ and ‘c’ affect the location of the vertex of the parabola.
Vertex Form of a Parabola
The vertex form is y = a(x – h)^2 + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify the vertex and the direction in which the parabola opens.
Examples of Parabola Equations
Example 1: Consider y = 2x^2 – 4x + 1. This is a parabola in standard form, opening upwards since ‘a’ (2) is positive.
Example 2: For y = -3(x + 2)^2 + 5, the vertex form indicates that the vertex is at (-2, 5), and the parabola opens downwards as ‘a’ (-3) is negative.
Converting Between Forms
Converting a parabola equation between different forms involves completing the square or rearranging terms. This conversion is useful for graphing or solving problems related to parabolas.
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