1. Parametric Equations & Plane Curves

2. These are called parametric equations.
There are times when we need to describe motion (or a curve) that is not a function.
We can do this by writing equations for the x and y coordinates in terms of a third variable (usually time or ).
These are called
parametric equations.
“t” is the parameter. (It is also the independent variable)

3. Let x = f(t) and y = g(t), where f and g are two functions whose common domain is some interval I. The collection of points defined by
(x, y) = (f(t), g(t))
is called a plane curve. The equations
x = f(t) y = g(t)
where t is in I, are called parametric equations of the curve. The variable t is called a parameter.

4. The variable t (the parameter) often represents time.
Picture this like a particle moving along and we know its x position over time and its y position over time.
We can figure out each of these and plot them together to see the movement of the particle.
Here are some examples of parametric functions

5. Discuss the curve given by the parametric equations

7. Try This! Graph the curve represented by the equations
The rectangular equation is

9. Determining a Rectangular Equation for Given Parametric Equations “Eliminating the Parameter”
Solve either equation for t.
Then substitute that value of t into the other equation.
Calculate the restrictions on the variables x and y based on the restrictions on t.

10. Find Rectangular Equation
Find a rectangular equation equivalent to

11. Find the rectangular equation of the curve whose parametric equations are
Parabola with vertex at (0, 2) that opens to the right.

12. Graph the plane curve represented by the parametric equations1 2 3

13. Now, eliminate the parameter
Solve for the parameter t in one of equations (whichever one is easier).
Substitute for t in the other equation.
We recognize this as a parabola opening up. Since our domain for t started at 0, it is only the right half.

14. Graph the plane curve represented by the parametric equations

15. Now, eliminate the parameter
Now, eliminate the parameter. Based on our curve we'd expect to get the equation of an ellipse. 2 2 4 4
When you want to eliminate the parameter and you have trig functions, it is not easy to solve for t because it requires inverse functions. Instead you solve for cos t and sin t and substitute them in the Pythagorean Identity:

16. If we let t = the angle on the unit circle, then:
Since:
We could identify the parametric equations as a circle.

17. You try! Write the rectangular equation for these parametric equations
This is the equation of an ellipse.

19. Adam Johnson throws a tennis ball with an initial speed of 40 meters per second at an angle of 45 degrees to the horizontal. The ball leaves Adam Johnson’s hand at a height of 2 meters.
(a) Find parametric equations that describe the position of the ball as a function of time.

20. (b) How long was the ball in the air?
When the ball hits the ground y(t)=0.
Solve as quadratic equation.

21. The ball was in the air for about 5.8 seconds.
Not possible.
The ball was in the air for about 5.8 seconds.

22. (c) When was the ball at its maximum height?
When y(t) has its maximum.
(d) What was the maximum height of the ball?
The maximum height is meters at t=2.89 seconds.

23. Find the parametric equations for the equation
Let y = t.