Glossary
Axis of Symmetry (Parabola)
A line that divides the parabola into two mirror-image halves.
Example:
For a parabola opening upwards, the vertical line passing through its vertex is its axis of symmetry.
Center (of a conic section)
The central point (h, k) from which an ellipse, circle, or hyperbola is symmetrically defined.
Example:
For an elliptical race track, the center is the midpoint of the track.
Circle
A set of all points equidistant from a central point, formed when a plane intersects a cone parallel to its base.
Example:
The boundary of a perfectly round pizza is a circle.
Conic Sections
Shapes formed by the intersection of a plane with a double-napped cone. These include circles, ellipses, parabolas, and hyperbolas.
Example:
The path of a satellite orbiting Earth can often be modeled as a conic section.
Directrix (of a parabola)
A fixed line used in the definition of a parabola; all points on the parabola are equidistant from this line and the focus.
Example:
Imagine a line on the ground; if you walk so you're always the same distance from a specific tree and that line, you're tracing a directrix for a parabolic path.
Ellipse
A closed curve where the sum of the distances from any point on the curve to two fixed points (foci) is constant.
Example:
The orbit of Mars around the Sun is an ellipse.
Foci (of an ellipse)
Two fixed points inside an ellipse such that the sum of the distances from any point on the ellipse to these two points is constant.
Example:
In a whispering gallery, sounds originating at one focus can be heard clearly at the other focus.
Focus (of a parabola)
A fixed point used in the definition of a parabola; all points on the parabola are equidistant from this point and the directrix.
Example:
A satellite dish is designed so that incoming signals reflect to the focus point, where the receiver is located.
Hyperbola
A curve consisting of two separate branches, formed when a plane intersects both parts of a double cone.
Example:
The path of a spacecraft using a gravitational slingshot maneuver can be approximated as a hyperbola.
Parabola
A U-shaped curve where every point is equidistant from a fixed point (focus) and a fixed line (directrix).
Example:
The trajectory of a basketball shot follows a parabola.
Radius (of a circle)
The constant distance from the center to any point on the circle.
Example:
If a circle has a diameter of 10 cm, its radius is 5 cm.
Standard Equation (Circle)
The algebraic form (x-h)² + (y-k)² = r² used to represent a circle with center (h,k) and radius r.
Example:
The equation (x - 5)² + (y + 1)² = 16 is the standard equation (circle) for a circle with center (5, -1) and radius 4.
Standard Equation (Ellipse)
The algebraic form (x - h)² / a² + (y - k)² / b² = 1 used to represent an ellipse.
Example:
The equation (x - 0)² / 25 + (y - 0)² / 9 = 1 is the standard equation (ellipse) for an ellipse centered at the origin.
Standard Equation (Hyperbola - opens left/right)
The algebraic form (x − h)² / a² - (y − k)² / b² = 1 used to represent a hyperbola that opens horizontally.
Example:
The equation (x - 1)² / 9 - (y - 2)² / 4 = 1 is a standard equation (hyperbola - opens left/right), indicating the hyperbola opens along the x-axis.
Standard Equation (Hyperbola - opens up/down)
The algebraic form -(x − h)² / a² + (y − k)² / b² = 1 used to represent a hyperbola that opens vertically.
Example:
The equation -(x + 3)² / 16 + (y - 4)² / 25 = 1 is a standard equation (hyperbola - opens up/down), showing the hyperbola opens along the y-axis.
Standard Equation (Parabola - opens left/right)
The algebraic form (y − k)² = a(x − h) used to represent a parabola that opens horizontally.
Example:
The equation (y - 1)² = 4(x + 2) describes a standard equation (parabola - opens left/right) with its vertex at (-2, 1).
Standard Equation (Parabola - opens up/down)
The algebraic form a(y − k) = (x − h)² used to represent a parabola that opens vertically.
Example:
The equation 2(y - 3) = (x - 0)² represents a standard equation (parabola - opens up/down) that opens upwards from its vertex at (0, 3).
Vertex (of a parabola)
The turning point of a parabola, located halfway between the focus and the directrix.
Example:
The lowest point of a suspension bridge cable, which often forms a parabolic shape, is its vertex.