Parametric equations describe curves using two functions, one for the x-coordinates and one for the y-coordinates, rather than a single function relating x and y. They allow curves to be described that cannot be expressed as a single-valued function. The parameter, often representing time or an angle, does not appear in the final graph but is used to generate the coordinates.

1. Parametric Equations

2. Introduction
Some curves in the plane can be described as
functions.
y = f (x)

3. Others...
cannot be described as functions.

4. Ways to Describe a Curve in the
Plane
An equation in two variables
Example: x + y − 2 x − 6 y + 8 = 0
2 2
This equation
describes a
circle.

5. A Polar Equation
r =θ
This polar equation
describes a double spiral.
We’ll study polar
curves later.

6. Parametric Equations
Example: x = t − 2t
2
y = t +1
The “parameter’’ is t.
It does not appear in the graph of the
curve!

7. Why?
The x coordinates of points on the curve
are given by a function.
x = t − 2t
2
The y coordinates of points on the
curve are given by a function.
y = t +1

8. Two Functions, One Curve?
Yes. If x = t − 2t and y = t + 1
2
then in the xy-plane the curve looks like
this, for values of t from 0 to 10...

10. Why use parametric equations?
• Use them to describe curves in the plane
when one function won’t do.
• Use them to describe paths.

11. Paths?
A path is a curve, together with a journey
traced along the curve.

12. Huh?
When we write
x = t − 2t
2
y = t +1
we might think of x as the x-coordinate
of the position on the path at time t
and y as the y-coordinate
of the position on the path at time t.

13. From that point of view...
The path described by
x = t − 2t
2
y = t +1
is a particular route along the curve.

15. As t increases
from 0, x first
decreases,
then increases. Path moves right!
Path moves left!

16. More Paths
To designate one route around the unit
circle use
x = cos(t )
y = sin(t )

17. That Takes Us...
counterclockwise from (1,0).

18. Where do you get that?
Think of t as an angle.
If it starts at zero, and increases to 2π ,
then the path starts at t=0, where
x = cos(0) = 1, and y = sin(0) = 0.

19. To start at (0,1)...
Use
x = sin(t )
y = cos(t )

20. That Gives Us...

21. How Do You Find The Path
• Plot points for various values of t, being
careful to notice what range of values t
should assume
• Eliminate the parameter and find one
equation relating x and y
• Use the TI82/83 in parametric mode

22. Plotting Points
• Note the direction the path takes
• Use calculus to find
– maximum points
– minimum points
– points where the path changes direction
• Example: Consider the curve given by
x = t + 1, y = 2t , − 5 ≤ t ≤ 5
2

23. Consider
x = t 2 + 1, y = 2t , − 5 ≤ t ≤ 5
• The parameter t ranges from -5 to 5 so the
first point on the path is (26, -10) and the
last point on the path is (26, 10)
• x decreases on the t interval (-5,0) and
increases on the t interval (0,5). (How can
we tell that?)
• y is increasing on the entire t interval (-5,5).
(How can we tell that?)

24. Note Further
x = t + 1, y = 2t , − 5 ≤ t ≤ 5
2
• x has a minimum when t=0 so the point
farthest to the left on the path is (1,0).
• x is maximal at the endpoints of the interval
[-5,5], so the points on the path farthest to
the right are the starting and ending points,
(26, -10) and (26,10).
• The lowest point on the path is (26,-10) and
the highest point is (26,10).

25. Eliminate the Parameter
Still use x = t + 1, y = 2t , − 5 ≤ t ≤ 5
2
Solve one of the equations for t
Here we get t=y/2
Substitute into the other equation
Here we get
x = ( y / 2) + 1 or x = ( y / 4) + 1
2 2

26. Summary
• Use parametric equations for a curve not
given by a function.
• Use parametric equations to describe paths.
• Each coordinate requires one function.
• The parameter may be time, angle, or
something else altogether...