Glossary
Cardioid
A heart-shaped polar curve, often described by equations of the form r = a ± b cos(θ) or r = a ± b sin(θ) where a = b.
Example:
The polar equation r = 1 + cos(θ) generates a cardioid shape.
Cartesian Plane
A two-dimensional coordinate system where points are located by their horizontal (x) and vertical (y) distances from the origin.
Example:
A straight line like y = 2x + 1 is typically graphed on the Cartesian plane.
Circle (Polar)
A set of points equidistant from a central point, which can be represented by specific polar equations involving sine or cosine.
Example:
The equation r = 4sin(θ) graphs a circle (polar) that passes through the origin.
Lemniscate
A figure-eight shaped polar curve, typically represented by equations like r² = a²cos(2θ) or r² = a²sin(2θ).
Example:
The equation r² = 9sin(2θ) produces a lemniscate graph.
Negative 'r' values
In polar coordinates, a negative 'r' value means plotting the point by moving in the opposite direction of the given angle θ from the origin.
Example:
The point (-3, π/4) is plotted by going 3 units in the direction of 5π/4 (π/4 + π) from the origin, demonstrating the effect of negative 'r' values.
Periodicity (Polar Functions)
The property of a polar function where its graph repeats itself after a fixed interval of the angle θ.
Example:
The function r = sin(2θ) has a periodicity (polar functions) of π, meaning its graph completes a full cycle every π radians.
Polar Coordinates
A system for locating points using an ordered pair (r, θ), where 'r' is the distance from the origin and 'θ' is the angle.
Example:
The point (3, π/2) represents a location 3 units from the origin along the positive y-axis in polar coordinates.
Polar Functions
Equations of the form r = f(θ) that describe curves using a distance from the origin (r) and an angle from the positive x-axis (θ).
Example:
The equation r = 2 + cos(θ) is a polar function that graphs a cardioid.
Polar Plane
A coordinate system where points are located by a distance from a central point (the origin) and an angle from a reference direction.
Example:
When plotting r = θ, you are drawing a spiral on the polar plane.
Spiral (Polar)
A curve that winds outwards or inwards from a central point, often generated by polar equations where 'r' is directly proportional to 'θ'.
Example:
The polar equation r = θ creates a spiral (polar) shape as θ increases.
Symmetry (Polar Graphs)
A property of a polar graph where it remains unchanged after certain transformations, such as rotation about the origin or reflection across an axis.
Example:
The graph of r = 2cos(θ) exhibits symmetry (polar graphs) about the x-axis.
r (radius)
In polar coordinates, 'r' represents the directed distance from the origin to a point.
Example:
In the polar coordinate (5, π/4), the r (radius) value is 5, indicating the point is 5 units from the origin.
θ (theta)
In polar coordinates, 'θ' represents the angle measured counterclockwise from the positive x-axis to the line segment connecting the origin to the point.
Example:
For the point (2, π/3), the θ (theta) value is π/3, indicating the angle from the positive x-axis.