Parametric equations are a mathematical representation of a curve or surface in terms of one or more independent parameters. These equations allow us to express the coordinates of points on the curve or surface in terms of the parameters.
To convert a Cartesian equation, such as x^2-y^2=1, to parametric equations, we can let x=cos(t) and y=sin(t), where t is the parameter. This will result in the parametric equations x=cos(t) and y=sin(t).
The parameter in parametric equations represents a specific point on the curve or surface. By varying the parameter, we can generate different points on the curve or surface, allowing us to graph and analyze it in a more efficient manner.
To graph parametric equations, first choose a range of values for the parameter. Then, plug in these values to the parametric equations to get corresponding x and y coordinates. Plot these points on a graph and connect them to form the curve or surface represented by the equations.
Parametric equations and polar coordinates are both ways to represent points on a curve or surface. While parametric equations use a parameter to describe the points, polar coordinates use an angle and a distance from the origin. Both systems can be converted to each other, making them interchangeable in some cases.