In Exercises 43-54 , find the standard form of the equation of the parabola with the given chara

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In Exercises 43-54 , find the standard form of the equation...

Key Concepts

Parabola

A parabola is a conic section defined as the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. Its geometric properties allow it to model quadratic relationships and it is fundamental in both algebra and geometry.

Vertex

The vertex is the point on the parabola that is closest to the focus and directrix. It serves as the point of symmetry and is often used as the origin in standard forms of the parabola, simplifying the equation and analysis of its properties.

Focus

The focus is a fixed point used in the definition of the parabola. Its position relative to the vertex determines the direction in which the parabola opens. When the vertex is the origin, the focus’s distance from the vertex, denoted by the parameter 'p', is crucial in forming the parabola's standard equation.

Parameter p

The parameter p represents the distance from the vertex to the focus (or directrix) and determines the 'width' or 'steepness' of the parabola. The value of p influences the coefficient in the standard equation that relates the squared term and the linear term.

Standard Form Equation

For a parabola with its vertex at the origin, the standard form of the equation depends on its orientation. For a horizontally opening parabola, the equation is written as y² = 4px, and for a vertically opening parabola, it is x² = 4py. This form directly incorporates the parameter p, relating the geometric definition to an algebraic expression.

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