10.6 Parametric Equations

Definition of Parametric Equations

parametric equation is a method of defining a relation using parameters. A simple kinematic1 example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion.

1. Kinematic.The branch of mechanics that studies the motion of a body or a system of bodies without consideration given to its mass or the forces acting on it.

http://en.wikipedia.org/wiki/Parametric_equation

We will not be finding parametric equations, in this example.

In this class we will graph parametric equation also eliminating the parameter.

Given the equation With respect to time

Rectangular equation Parametric Equations

The parametric equations have to be Continuous

xx

y

72

2

tty

tx

22416

2242

Both Equations graph the same. The Parametric states how fast to graph

Given the equation With respect to time

Rectangular equation Parametric Equations

xx

y

72

2

tty

tx

22416

2242

0,72

2

23t

4

23t

18,36

Graphing Parameter equations

Parameter equations

2

4

1

ty

tx

t x y-4 -1 16-2 -0.5 40 0 02 0.5 44 1 16

Graphing Parameter equations

Parameter equations

2

4

1

ty

tx

t x y-4 -1 16-2 -0.5 40 0 02 0.5 44 1 16

Change to Rectangular Equation

Parameter equations to

Rectangular equation

t x y-4 -1 16-2 -0.5 40 0 02 0.5 44 1 16

2

22

16

4

44

1

xy

xyty

txtx

Change from Parametric to Rectangular

Parametric Equations

121 xxtxx 121 yytyy

Change from Parametric to Rectangular

Parametric Equations

txx

xx

xxtxx

12

1

121 121 yytyy

Change from Parametric to Rectangular

Parametric Equations

txx

xx

xxtxx

12

1

121

12

12

11

121

yyxx

xxyy

yytyy

Change from Parametric to Rectangular

Parametric Equations

txx

xx

xxtxx

12

1

121

12

1211

12

12

11

121

xx

yyxxyy

yyxx

xxyy

yytyy

Change from Parametric to Rectangular

Parametric Equations

txx

xx

xxtxx

12

1

121

mxxyy

xx

yyxxyy

yytyy

11

12

1211

121

Change from Parametric to Rectangular

Parametric Equations

txx

xx

xxtxx

12

1

121

11

12

1211

121

xxmyy

xx

yyxxyy

yytyy

Change from Parametric to Rectangular

x = h + a Cosθ y = k + b Sinθ

Cosa

hx

Sinb

ky

Change from Parametric to Rectangular

x = h + a Cosθ y = k + b Sinθ

Cosa

hx

Sinb

ky

1

1

1

2

2

2

2

22

22

b

ky

a

hx

b

ky

a

hx

SinCosSince

What would the parametric equation be for a circle with center at (3,2) and radius of 4

(x – h)2 + (y – k)2 = r2 In a circle a = b = 4

Cosr

hx

Sinr

ky

1

2

2

2

2

r

ky

r

hx

SinySiny

CosxCosx

424

2

434

3

Homework

Page 747 – 749

# 7, 14, 21, 28,

32, 41, 48

Homework

Page 747 – 749

# 5, 10, 15, 20,

25, 30, 35, 40