10.5 Parametric Equations

Parametric equations

• A third variable t (a parameter) tells us when an object is at a given point (x, y)

• Both x and y are functions of time (t)

• If f and g are continuous functions of t on an interval I then x = f(t) and y = g(t)

• Different parametric equations can be used to represent various speeds at which an object travels a given path

On the calculator

• Change MODE to PAR

• In y= you will now have X1t and Y1t

• Use the window to set the max and min time values as well as the max and min x and y values

Eliminating the parameter

• Solve one of the parametric equations for t

• Substitute into the other equation for t

• The rectangular equation should contain x and y variables only

• Converting from parametric to rectangular can change the range of x and y so you may need to restrict your range in the rectangular equation

Find the rectangular equation for

ty

tx

1

tx 221 xy

Parametric Rectangular

Since the parametric is defined only when t ≥0 and the x is always positive, you must restrict the domain of the rectangular so that x≥0

Angle parameters• Solve for the trig function and use a

trig identity to substitute in the values and form an equation in x and y

• Example: x = cosθ

y = 3 sin θ sin θ= y/3

sin2θ + cos2θ=1

(y/3)2 + x2 = 1 119

22

xy

Ellipse with center (0, 0)

Exponential parameters

• Recall that ex and ln x are inverse functions• Example: x = ln 2t

y = 2t2

ex = eln2t

ex = 2t

t= ex/2

2

22

xey

xey 2

2

1