Section 6.3 Parametric Equations

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Presentation transcript:

Section 6.3 Parametric Equations

Butterfly Curve http://www.youtube.com/watch?v=7PC5ITad8DE Try it yourself! Change your mode to PAR

6.3 Parametric Equations Graph parametric equations. Determine an equivalent rectangular equation for parametric equations. Determine parametric equations for a rectangular equation.

Graphing Parametric Equations We have graphed plane curves that are composed of sets of ordered pairs (x, y) in the rectangular coordinate plane. Now we discuss a way to represent plane curves in which x and y are functions of a third variable t. One method will be to construct a table in which we choose values of t and then determine the values of x and y.

Example 1 Graph Parametric Graph the curve represented by the equations The rectangular equation is

Parametric Equations If f and g are continuous functions of t on an interval I, then the set of ordered pairs (x, y) such that x = f(t) and y = g(t) is a plane curve. The equations x = f(t) and y = g(t) are parametric equations for the curve. The variable t is the parameter.

Determining a Rectangular Equation for Given Parametric Equations Solve either equation for t. Then substitute that value of t into the other equation. Calculate the restrictions on the variables x and y based on the restrictions on t.

Example 2 Find Rectangular Find a rectangular equation equivalent to Solution The rectangular equation is:

Determining Parametric Equations for a Given Rectangular Equation Many sets of parametric equations can represent the same plane curve. In fact, there are infinitely many such equations. The most simple case is to let either x (or y) equal t and then determine y (or x).

Example 4 Find Parametric Equations Find three sets of parametric equations for the parabola Solution