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Definition

A parabola is a symmetric, U-shaped curve defined as the set of all points in a plane equidistant from a given point called the focus and a given line called the directrix. The standard form of its equation is $y = ax^2 + bx + c$.

5 Must Know Facts For Your Next Test

1. The vertex of a parabola is the highest or lowest point on the graph, depending on its orientation.
2. The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror images.
3. The focus lies inside the parabola, while the directrix lies outside it; both help define its shape.
4. If 'a' in $y = ax^2 + bx + c$ is positive, the parabola opens upwards; if negative, it opens downwards.
5. The distance between any point on the parabola and its focus equals the perpendicular distance from that point to the directrix.

Review Questions

• What defines the shape and position of a parabola?
• How do you determine whether a parabola opens upwards or downwards?
• What are the roles of focus and directrix in defining a parabola?

Related terms

Vertex: The highest or lowest point on a parabolic graph where it changes direction.

Axis of Symmetry: A vertical line through the vertex that divides the parabola into two symmetrical halves.

Focus: A fixed point used with the directrix to define a parabola; all points on the curve are equidistant from this point and the directrix.