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

step 1 This is 62 number question. The problem is, the equation is x minus 5 plus y plus 4 square equal

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

Key Concepts

Parabola

A parabola is a conic section defined as the set of points equidistant from a fixed point called the focus and a fixed line called the directrix. It is a fundamental curve in algebra and geometry, often represented in a standard equation form that makes it easier to identify its key features and graph it.

Standard Form of a Parabola

Writing a parabola's equation in standard form reveals its geometric properties such as the vertex, axis of symmetry, and the parameter that determines how “wide” or “narrow” it is. The standard form, either (y - k)^2 = 4p (x - h) or (x - h)^2 = 4p (y - k), clearly shows the vertex (h, k) and indicates the orientation of the parabola based on the squared variable.

Vertex

The vertex of a parabola is the point where the curve changes direction. It acts as the turning point and is a crucial element for graphing, as it serves as a reference for locating other features like the focus and directrix. In the standard form of a parabola, the vertex is immediately identifiable.

Focus

The focus of a parabola is a fixed point used in its geometric definition. Every point on the parabola is equidistant from the focus and the directrix. The location of the focus relative to the vertex is determined by the parameter p in the standard form equation, and it plays a key role in understanding the reflective properties of the parabola.

Directrix

The directrix is a fixed line used in defining a parabola. Similar to the focus, it is part of the condition that all points on the parabola are equidistant from both the focus and the directrix. The position of the directrix is determined by the same parameter p that locates the focus, and it influences the overall orientation of the parabola.

Graphing a Parabola

Graphing a parabola involves identifying its vertex, axis of symmetry, focus, and directrix, and understanding its orientation (whether it opens upward, downward, left, or right). Once these components are determined from the standard form of the equation, accurate sketches can be made, illustrating both the shape of the parabola and its geometric properties.

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