A parametric curve is a mathematical concept that represents a curve in terms of one or more parameters. These parameters are typically represented by variables such as t or u, and they determine the coordinates of points along the curve.
To sketch a parametric curve, you will need to plot points using the coordinates determined by the parameter(s). Then, connect the points to create a smooth curve. You can also use a graphing calculator or computer software to plot the curve.
A Cartesian equation is an equation that describes a curve or shape in terms of x and y coordinates. It is written in the form y = f(x), where f(x) is a function of x. This type of equation is used to represent curves that are not easily described by a single equation.
To find the Cartesian equation of a parametric curve, you will need to eliminate the parameter(s) from the equations. This can be done by solving for the parameter(s) in terms of x and y and substituting them into the original equations. The resulting equation will be the Cartesian equation of the curve.
Parametric curves and Cartesian equations are used in various fields of science and engineering, including physics, astronomy, and computer graphics. They are also commonly used in designing and modeling complex shapes and curves in architecture and product design. Additionally, they are essential in understanding and analyzing motion and trajectories in kinematics and dynamics.