Precalculus Parametric Equations graphs

Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric equations.  Determine the location of a moving object at a specific time.

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. 822

Example 1 Graph Parametric Graph the curve represented by the equations Solve for x when t=-3 and t=3 Solve for y when t=-3 and t=3 These are the restrictions on x and y – now make a table for all values of t… 822

Example 1 Graph Parametric Graph the curve represented by the equations Find the rectangular equation by eliminating t: First, solve for t in the first equation 822

Example 1 Graph Parametric Graph the curve represented by the equations t = 2x – substitute 2x for t in the second equation… 822

Example 1 Graph Parametric Graph the curve represented by the equations The rectangular equation is: 822

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. 823

Determining a Direct Relationship 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. 823

Example 2 Find Rectangular Find a rectangular equation equivalent to 823

Example 2 Find Rectangular Find a rectangular equation equivalent to Solution The rectangular equation is: 823

Example 3 X = 2cos t Y = 2sin t Square both equations and add together (This one we will handle differently)

Example 3 X = 2cos t Y = 2sin t Add the squares…. A circle of radius 2 = 1

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 ). 826

Example 4 Find Parametric Equations Find three sets of parametric equations for the parabola Solution 826

Parametization of a line Find the parametric equations going through the points, p (-2,2) and q (3,6) Step 1: Find the vector Step 2: Find the vector from p to (x,y) Step 3: set up:

solution Set x’s and y’s equal, separately: X+2 = 5t Y-2 = 4t Solve for x and y: X = 5t – 2 Y = 4t + 2