9.4 Parametric Equations
Single equations have always been solved involving two variables, x and y. A third variable can be introduced to determine time and add direction, the variable t is called a parameter, and it is possible to write both x and y as functions of t to obtain parametric equations. Motion is also given, resulting in the graph of a plane curve.
If f and g are continuous functions of t on an interval l, then the set of ordered pairs (f (t), g(t))
is a plane curve C. The equations given by
x = f(t) and y = g(t)
are parametric equations for C, and t is the parameter.
Graphs of Plane Curves
Curves can be represented by a pair of parametric equations is to plot points in the xy plane. Each set of coordinates (x, y) is determined from a value chosen for the parameter t. By plotting the resulting points in the order of increasing values of t, the curve is traced in a specific direction, called the orientation of the curve.
If f and g are continuous functions of t on an interval l, then the set of ordered pairs (f (t), g(t))
is a plane curve C. The equations given by
x = f(t) and y = g(t)
are parametric equations for C, and t is the parameter.
Graphs of Plane Curves
Curves can be represented by a pair of parametric equations is to plot points in the xy plane. Each set of coordinates (x, y) is determined from a value chosen for the parameter t. By plotting the resulting points in the order of increasing values of t, the curve is traced in a specific direction, called the orientation of the curve.
Sketching a Plane Curve
More than one set of equations can represent a single graph, yet the t values are different, so the one with the smaller amount of t, means the particle moved more rapidly.
Eliminating the Parameter
Many curves that are represented by sets of parametric equations have graphs that can also be represented by rectangular equations. The process of finding the rectangular equation is called eliminating the parameter. This is typically solving for t and then substituting in, but sometimes it is beneficial to use Pythagorean identities to eliminate the parameter.
Many curves that are represented by sets of parametric equations have graphs that can also be represented by rectangular equations. The process of finding the rectangular equation is called eliminating the parameter. This is typically solving for t and then substituting in, but sometimes it is beneficial to use Pythagorean identities to eliminate the parameter.
Eliminating the Parameter
Finding Parametric Equations for a Graph
Doing the Reverse.
Doing the Reverse.
Finding Parametric Equations for a Given Graph