Parametric Representation of Curves

Lesson 9.4

Parametric Representation

• Define variables (x,y) to be each functions of some other variable– usually t for time

• Sox = g(t) and y = h(t)

• The calculator has a parametric mode

Parametric Mode on Calculator

• Note the appearance ofthe Y= screen– must use t– must have both a function

for x and for y

• Note also thechange in the window specs

Eliminate the Parameter

• Use substitution

• Solve for t

• Set results equalto each other

• Solve for y

3 2

3

23

23 2

x t y t

yt x t

yx

y x

Eliminate the Parameter

• Try this one

2

2

2

2

x t

y t

Eliminate the Parameter

• Use Trig Identities

• What is this figure?

2 2

2 2

4 sin 3 cos

sin 4 cos 3

We know sin cos 1 so ...

(4 ) ( 3) 1

x t y t t

t x t y

t t

x y

Eliminate the Parameter

• Use other relationships:– consider

y = ln x

12 2 t tx y ln ln 2 ln (1 ) ln 2

ln ln 1

ln 2 ln 2ln ln

1ln 2 ln 2ln ln 2 ln

2ln ln

2

x t y t

x yt t

x y

x y

xy

xy

Finding Derivatives

• Given x = g(t) y = h(t)• Then

• Try

'( )

'( )

dydy h tdt

dxdx g tdt

3 2 2 4x t y t

Finding Derivatives

• For

… we get

• To evaluate, substitute a specific t in

• Also possible to eliminate the parameter with substitution

3 2 2 4x t y t

23 3

4 4

dy t t

dx t

Area under the Parametric Curve

• Given x = x(t) y = y(t)

• Thena=x(t1) b=x(t2)

2

1

( ) ( ) '( )ta

b t

y x dx y t x t dt

Area under the Parametric Curve

• Try this:2 23 ( 1)

0 1

x t y t

t

Assignment

• Lesson 9.4

• Page 659

• Exercises:5, 9, 13, 15, 19, 23, 25, 29, 31, 33, 35, 37, 41