Glossary
Cartesian Equation
An equation that expresses a relationship between variables, typically x and y, directly without the use of a third parameter.
Example:
After eliminating the parameter from x(t) = 2t and y(t) = 4t², the resulting Cartesian equation y = x² describes the same parabolic path.
Domain (of a parametric function)
The specified range of values for the parameter 't' that defines the extent or segment of the parametric curve.
Example:
If a particle's motion is described by x(t) = t and y(t) = t² for 0 ≤ t ≤ 5, then [0, 5] is the domain of the parametric function, limiting the path.
Eliminate the parameter
The algebraic process of converting a set of parametric equations (x=f(t), y=g(t)) into a single Cartesian equation relating x and y directly.
Example:
To eliminate the parameter from x(t) = t+2 and y(t) = t², one could solve the first equation for t (t=x-2) and substitute it into the second, yielding y = (x-2)².
Parameter (t)
The independent third variable, typically denoted as 't', used in parametric equations to define both the x and y coordinates of a point on a curve.
Example:
In the equations x(t) = 5t and y(t) = t² + 1, t is the parameter that dictates the position of a point at any given 'time' or value of t.
Parametric Equations of Circles
A specific set of parametric equations, typically x(t) = h + r cos(t) and y(t) = k + r sin(t), that describe a circle with center (h, k) and radius r.
Example:
The parametric equations of circles x(t) = 3 + 5cos(t) and y(t) = -2 + 5sin(t) describe a circle centered at (3, -2) with a radius of 5.
Parametric Equations of Ellipses
A specific set of parametric equations, typically x(t) = h + a cos(t) and y(t) = k + b sin(t), that describe an ellipse with center (h, k) and semi-axes a and b.
Example:
The parametric equations of ellipses x(t) = 1 + 4cos(t) and y(t) = 2 + 3sin(t) describe an ellipse centered at (1, 2) with a horizontal semi-axis of 4 and a vertical semi-axis of 3.
Parametric Functions
A way to describe curves and surfaces in a 2D space using a set of equations where both x and y coordinates are defined by a third variable, called the parameter.
Example:
When modeling the path of a projectile, its horizontal position x(t) and vertical position y(t) are often described using parametric functions of time t.
Parametric Representation
The set of equations, x = f(t) and y = g(t), that collectively define a curve using a parameter 't'.
Example:
The equations x(t) = cos(t) and y(t) = sin(t) provide a parametric representation of a unit circle.
Start/End Points
The specific coordinates (x, y) on a parametric curve that correspond to the minimum and maximum values of the parameter's domain.
Example:
For x(t) = t+1, y(t) = t-1 with 0 ≤ t ≤ 3, the start/end points would be (1, -1) when t=0 and (4, 2) when t=3.
Table of Values
A systematic list of corresponding x and y coordinates generated by evaluating parametric equations for different values of the parameter 't'.
Example:
To graph x(t) = t-1 and y(t) = t², a student might create a table of values for t = -2, -1, 0, 1, 2 to plot points like (-3, 4), (-2, 1), (-1, 0), (0, 1), (1, 4).