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

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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 known as the directrix. It has a characteristic U-shape and can open upward, downward, leftward, or rightward depending on its orientation.

Standard Form of a Parabola

The standard form of a parabola's equation allows one to easily identify its key geometric features including the vertex, focus, and directrix. For parabolas that open horizontally, the equation is typically written as x = a(y - k)^2 + h, where (h, k) is the vertex of the parabola.

Completing the Square

Completing the square is an algebraic method used to rewrite quadratic expressions in a perfect square form. This technique is crucial for converting a quadratic equation into standard form, which in turn facilitates the identification of the parabola’s vertex and other properties.

Vertex

The vertex of a parabola is the point where the curve changes direction; it is the point of maximum or minimum value on the graph depending on whether the parabola opens downward/upward or left/right. In the standard form of the equation, the vertex is represented by the coordinates (h, k).

Focus

The focus of a parabola is a fixed point used in its definition, located along the axis of symmetry. The distance from the vertex to the focus is determined by the value of the parameter in the standard equation, and it plays a key role in the parabola's reflective properties.

Directrix

The directrix of a parabola is a fixed line that, together with the focus, defines the set of points on the parabola. Every point on the parabola is equidistant from the focus and the directrix, and its equation can be derived once the parabola is written in standard form.

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