1. Parametric Equations

2. Cartesian Equations
Equations defined in terms of x and y. These may or may not be functions. Some examples include:
x2 + y2 = 4
y = x2 + 3x + 2

3. Parametric Equations
Equations where x and y are functions of a third variable, such as t. That is,
x = f(t) and y = g(t).
The graph of parametric equations are called parametric curves and are defined by (x, y) = (f(t), g(t)).

4. Example
The path of a particle in two-dimensional space can be modeled by the parametric equations x = 2 + cos t and y = 3 + sin t. Sketch a graph of the path of the particle for 0 ≤ t ≤ 2p.

5. t
x = 2 + cos(t)
y = 3 + sin (t)
0.00
3.00
0.52
2.87
3.50
1.05
2.50
3.87
1.57
2.00
4.00
2.09
1.50
2.62
1.13
3.14
1.00
3.67
4.19
2.13
4.71
5.24
5.76
6.28

6. Plot of x = 2 + cos t and y = 3 + sin t
How is t represented on this graph?

7. Plot of x = 2 + cos t and y = 3 + sin t
 t = 
 t = 0 

8. Parametric Equations and Technology
Graphing calculators and mathematical software can plot parametric equations much more efficiently then we can, so use either and plot the following equations. In what direction is t increasing?
(a) x = t2, y = t3 (b)
(c) x = sec θ, y = tan θ; -/2 < θ < /2

9. Converting Equations
Parametric equations can easily be converted to Cartesian equations by solving one of the equations for t and substituting the result into the other equation.

10. You try it. Hint: sec2 θ – tan2 θ = 1
You try it for x = sec θ, y = tan θ where
Hint: sec2 θ – tan2 θ = 1

11. Cycloids
A cycloid is the graph of the path of a fixed point P on a circle of radius r that rolls along a straight line.
x = r( – sin )
y = r(1 – cos )