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

step 1 This is 71 number problem. Y square plus X plus Y equal to 0, which means Y square plus Y plus X

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

Key Concepts

Parabola

A parabola is a conic section defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). This definition is central in understanding its geometric properties and how its algebraic equation represents these characteristics.

Completing the Square

Completing the square is an algebraic method used to rewrite quadratic equations into a form that more clearly reveals the vertex of the parabola. This process rearranges and groups terms to form a perfect square plus or minus a constant, making it easier to analyze and graph the conic section.

Vertex Form of a Parabola

The vertex form of a parabola's equation, typically written as (y - k)^2 = 4p(x - h) or (x - h)^2 = 4p(y - k), directly shows the vertex at the point (h, k) and reveals the distance p between the vertex and the focus or directrix. This form is invaluable for graphing and identifying key features of the parabola.

Focus

The focus is a specific point associated with a parabola that, along with the directrix, defines the set of points on the curve. The distance between the vertex and the focus, often denoted as p, determines the 'sharpness' or 'flatness' of the parabola's graph.

Directrix

The directrix is a fixed line used in the definition of a parabola. It lies a distance p from the vertex on the side opposite the focus, ensuring that any point on the parabola is equidistant from the focus and the directrix. This property is essential in deriving the parabola's equation and understanding its geometry.

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