In Exercises 55-72 , find the vertex, focus, and directrix of the parabola and sketch its graph.

step 1 This is 65 number question x plus 3 by 2 whole square equal to 4 into y minus 2 so this is 4 4 i

In Exercises 55-72 , find the vertex, focus, and directrix...

Key Concepts

Parabola

A parabola is a type of conic section defined as the set of all points that are equidistant from a fixed point called the focus and a fixed line called the directrix. This geometric definition underlies its reflective properties and appearance in various physical phenomena and is fundamental in the study of quadratic functions.

Vertex

The vertex of a parabola is the point at which the curve changes direction. In its standard or vertex form, the location of the vertex is readily identifiable, and it serves as a reference point from which other features such as the focus and directrix are determined. Its coordinate is central to graphing the parabola accurately.

Focus

The focus of a parabola is a fixed point that, together with the directrix, defines the set of points constituting the parabola. It lies along the axis of symmetry and is located at a distance from the vertex based on the parabola's focal parameter. The focus is essential in establishing the parabola’s geometric properties and in various applications, including optics and engineering.

Directrix

The directrix of a parabola is a fixed line that is used, along with the focus, to define the curve. Every point on the parabola is equidistant to the focus and the directrix, making this line pivotal in the formal geometric definition. Understanding the directrix aids in the analysis and graphing of the parabola.

Standard Form (Vertex Form) of a Parabola

The vertex form of a parabola's equation is a reformulated version of the quadratic equation that clearly shows the vertex and facilitates the identification of other key features such as the focal length. This form is particularly useful for graphing because it directly reveals the translation of the parabola from the origin and the parameter that determines its opening and stretching.

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