Glossary
Conic Sections
Curves formed by the intersection of a plane with a double-napped cone, including circles, ellipses, parabolas, and hyperbolas, which often have standard parametric forms.
Example:
The orbits of planets around the sun are classic examples of conic sections, specifically ellipses, which can be precisely described using parametric equations.
Domain of 't'
The set of all possible values for the parameter 't' that define the specific portion or entire extent of a parametrized curve.
Example:
For a full circle, the domain of 't' is typically 0 ≤ t < 2π, ensuring one complete revolution without retracing.
Ellipses (Parametrization)
The method of defining an elliptical curve using parametric equations, typically involving cosine and sine functions for the x and y coordinates, respectively, centered at (h, k).
Example:
To simulate the motion of a point around an oval track, one might use ellipses (parametrization) with x(t) = h + a cos(t) and y(t) = k + b sin(t) to define its path.
Hyperbolas (Parametrization)
The technique of representing a hyperbolic curve using parametric equations, which commonly involve secant and tangent functions, depending on whether the hyperbola opens horizontally or vertically.
Example:
In physics, the path of a particle repelled by a force might follow a hyperbola, which can be described using hyperbolas (parametrization) to model its trajectory.
Implicit Functions
Functions where the dependent variable is not explicitly isolated on one side of the equation, often defining a relationship between x and y without directly stating y as a function of x.
Example:
The equation x² + y² = 25 is an implicit function that describes a circle, where y is not explicitly defined in terms of x.
Parabolas (Parametrization)
The process of representing a parabolic curve using parametric equations, often by setting one coordinate equal to 't' and expressing the other in terms of 't'.
Example:
A projectile's path, which is a parabola, can be described using parabolas (parametrization), where x(t) and y(t) depend on time t to show its horizontal and vertical positions.
Parameter ('t')
The independent variable used in parametric equations, which typically represents time, an angle, or another quantity that determines the position of a point on a curve.
Example:
In the parametric equations for a projectile's flight, the parameter 't' often represents the elapsed time in seconds, dictating the projectile's location.
Parametrization
A method of defining a curve using a single independent variable, typically 't', where the x and y coordinates are expressed as functions of 't'. This allows describing the curve's path over time or another changing quantity.
Example:
To describe the path of a satellite orbiting Earth, engineers use parametrization to define its x and y positions at any given time t.
Parametrizing Functions (y=f(x))
The process of converting a function explicitly defined as *y = f(x)* into parametric equations by setting *x(t) = t* and *y(t) = f(t)*.
Example:
To trace the graph of y = x² + 3 in a graphing calculator that accepts parametric input, one can use parametrizing functions by setting x(t) = t and y(t) = t² + 3.
Parametrizing Inverse Functions (x=f⁻¹(y))
The process of converting an inverse function *x = f⁻¹(y)* into parametric equations by setting *x(t) = f⁻¹(t)* and *y(t) = t*.
Example:
If you have a function like x = √y, you can use parametrizing inverse functions to represent it as x(t) = √t and y(t) = t, effectively swapping the roles of x and y.
Semi-major axis ('a')
For an ellipse, 'a' represents half the length of the longest diameter. For a hyperbola, it's the distance from the center to a vertex along the transverse axis.
Example:
In the parametric equation for an elliptical orbit, a larger value for the semi-major axis ('a') indicates a wider and longer orbit.
Semi-minor axis ('b')
For an ellipse, 'b' represents half the length of the shortest diameter, perpendicular to the semi-major axis. For a hyperbola, it relates to the conjugate axis.
Example:
A smaller value for the semi-minor axis ('b') in an ellipse makes the shape appear more elongated or 'squashed'.
Unit Circle
A circle centered at the origin with a radius of 1, commonly parametrized using trigonometric functions like *x(t) = cos(t)* and *y(t) = sin(t)*.
Example:
When studying trigonometry, the coordinates of points on the unit circle are often given by (cos θ, sin θ), which is a fundamental example of parametrization.