Section 11 1 Plane Curves and Parametric Equations

Section 11. 1 Plane Curves and Parametric Equations By Kayla Montgomery and Rosanny Reyes

Introduction We typically think of a graph as a curve in the xy-plane generated by the set of all ordered pairs of the form (x, y) = (x, f (x)) for a ≤ x ≤ b. In regular graphs some planes on a curve can be described as functions y = sinx

Introduction cont. Others cannot be described as functions Plane Curve – When x and y are continuous functions of t Irregular Plane Curve What is t? ? ? ? Wait and See!!!!

Parameters and Parametric Equations Parameter = t Third Variable determines when an object was at a given point (x, y) Parametric Equations Writing both x and y as functions of t

Sketching the Curve These new points (x, y) = (f(t), g(t)) In the plane are called the graph of the curve C These points are still plotted on the (x, y) plane Each set of coordinates are determined by a value chosen for the parameter t Plotting these points in order of increasing values of t is called the curve orientation

Example 1 t x y -2 -1 0 1 2 3 0 -3 -4 -3 0 5 -1 -½ 0 ½ 1 3/ 2

Example

Eliminating the Parameter Parametric Equations Solve for t in one equation Substiute into second equation t = 2 y Rectangular equation

Example

Adjusting the Domain After Eliminating the Parameter

Using a Trigonometric Identity to Eliminate a Parameter From this rectangular equation we see that the graph is an ellipse centered at (0, 0), with vertices at (0, 4) and (0, -4) and minor axis of length 2 b = 6