Section 11.1Section 11.1Plane Curves and Parametric Plane Curves and Parametric

EquationsEquations

By Kayla Montgomery By Kayla Montgomery

andand

Rosanny ReyesRosanny Reyes

IntroductionIntroduction

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 Others cannot be described as described as functionsfunctions

Plane Curve – When Plane Curve – When xx and and yy are continuous are continuous functions of functions of tt

What is What is tt??????????????

Wait and See!!!!Wait and See!!!!

Irregular

Plane Curve

Parameters and Parameters and Parametric EquationsParametric Equations

Parameter = Parameter = tt

Third Variable Third Variable determines when an determines when an object was at a given object was at a given point point (x,y)(x,y)

Parametric EquationsParametric Equations

Writing both x and y as Writing both x and y as functions of tfunctions of t

Sketching the CurveSketching the Curve

These new points These new points (x,y) = (f(t), g(t))(x,y) = (f(t), g(t))

In the plane are called the graph of the In the plane are called the graph of the curve curve CC

These points are still plotted on the (x,y) These points are still plotted on the (x,y) planeplane

Each set of coordinates are determined by a Each set of coordinates are determined by a value chosen for the parameter value chosen for the parameter tt

Plotting these points in order of increasing Plotting these points in order of increasing values of values of t t is called the curve orientationis called the curve orientation

Example 1Example 1

t -2 -1 0 1 2 3

x 0 -3 -4 -3 0 5

y -1 -½

0 ½ 1 3/2

Example Example

Eliminating the ParameterEliminating the Parameter

Parametric

EquationsSolve for t in

one equationSubstiute into

second equation

Rectangular

equation

t = 2y

ExampleExample

Adjusting the Domain After Adjusting the Domain After Eliminating the ParameterEliminating the Parameter

Using a Trigonometric Identity to Using a Trigonometric Identity to Eliminate a ParameterEliminate 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 2b = 6