key term - Parametric Equations

Definition

Parametric equations are a way of representing a curve or a surface in the coordinate plane or space by expressing the coordinates as functions of a parameter. This parameter is typically represented by a variable, such as $t$, that can take on different values to generate the points that make up the curve or surface.

5 Must Know Facts For Your Next Test

1. Parametric equations are often used to represent curves and surfaces that cannot be easily expressed using a single equation in $x$ and $y$ (or $x$, $y$, and $z$).
2. In the context of the unit circle, parametric equations are used to represent the sine and cosine functions, where the parameter $t$ is the angle in radians.
3. Parametric equations can be used to graph more complex curves and surfaces, such as cycloids, epicycloids, and hypocycloids.
4. Parametric equations can also be used to solve systems of linear equations with three variables, by expressing the variables in terms of a parameter.
5. The flexibility of parametric equations allows for the representation of curves and surfaces that cannot be easily expressed using traditional Cartesian coordinates.

Review Questions

• Explain how parametric equations are used to represent the sine and cosine functions in the context of the unit circle.
  • In the unit circle, the $x$-coordinate is given by the cosine function, $x = \cos(t)$, and the $y$-coordinate is given by the sine function, $y = \sin(t)$, where $t$ is the angle in radians. This parametric representation allows for the easy generation of points on the unit circle by varying the parameter $t$. The parametric equations $x = \cos(t)$ and $y = \sin(t)$ provide a flexible way to describe the circular motion of the unit circle and the periodic nature of the sine and cosine functions.
• Describe how parametric equations can be used to graph more complex curves and surfaces, such as cycloids, epicycloids, and hypocycloids.
  • Parametric equations allow for the representation of curves and surfaces that cannot be easily expressed using a single equation in $x$ and $y$ (or $x$, $y$, and $z$). For example, a cycloid is the curve traced by a point on the circumference of a circle as it rolls along a straight line. This curve can be represented using parametric equations, where the $x$-coordinate is given by $x = t - \sin(t)$ and the $y$-coordinate is given by $y = 1 - \cos(t)$, with the parameter $t$ representing the angle of rotation. Similarly, epicycloids and hypocycloids, which are curves generated by the motion of one circle rolling around the outside or inside of another circle, can be represented using parametric equations. This flexibility allows for the graphing of a wide range of complex curves and surfaces.
• Explain how parametric equations can be used to solve systems of linear equations with three variables.
  • Parametric equations can be used to solve systems of linear equations with three variables by expressing the variables in terms of a parameter. For example, a system of three linear equations with three variables, $x$, $y$, and $z$, can be represented using parametric equations, where $x$, $y$, and $z$ are expressed as functions of a parameter, such as $t$. This allows the system to be solved by finding the values of the parameters that satisfy the equations. The flexibility of parametric equations enables the representation of the variables in a way that can simplify the solution process for systems of linear equations with three variables, which can be challenging to solve using traditional methods.

Related terms

Parametric Representation: The representation of a curve or surface using parametric equations, where the coordinates are expressed as functions of a parameter.

Parametric Curve:

A curve that is defined by a set of parametric equations, where the coordinates are given as functions of a parameter.

Parametric Surface:

A surface that is defined by a set of parametric equations, where the coordinates are given as functions of two parameters.