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...