1. PARAMETRIC Q U A T I N S

2. The variable t (the parameter) often represents time
The variable t (the parameter) often represents time. We can picture this like a particle moving along and we know its x position over time and its y position over time and we figure out each of these and plot them together to see the movement of the particle.

3. t 1 2 3 Graph the plane curve represented by the parametric equations
The t values we pick must be greater than or equal to 0. Let's start with 0. t 1 2 3
We'll make a chart and choose some t values and find the corresponding x and y values.
We see the "path" of the particle. The orientation is the direction it would be moving over time (shown by the arrows)

4. We could take these parametric equations and find an equivalent rectangular equation with substitution. This is called "eliminating the parameter."
Solve for the parameter t in one of equations (whichever one is easier).
Substitute for t in the other equation. 2 2
We recognize this as a parabola opening up. Since our domain for t started at 0, it is only the right half.

5. t Graph the plane curve represented by the parametric equations
The t values we pick must be from 0 to 2  t
Make the orientation arrows based where the curve was as t increased.
You could fill in with more points to better see the curve.

6. Let's eliminate the parameter
Let's 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. Instead you solve for cos t and sin t and substitute them in the Pythagorean Identity:
Here is the rectangular version of our ellipse. You can see it matches!

7. Your Casio graphic calculator can plot parametric equations
Your Casio graphic calculator can plot parametric equations. Select “Graph” mode and check that “Type” is set to Parm.
When you then enter a graph, it will have t for the variable and you can enter more than one equation.
If you watch as it draws the graph, you will see the orientation (direction) of the curve.

8. If an object is dropped, thrown, launched etc
If an object is dropped, thrown, launched etc. at a certain angle and has gravity acting upon it, the equations for its position at time t can be written as:
horizontal position
initial velocity
angle measured from horizontal
gravitational constant which is 9.8 m/s2
time
vertical position
initial height

9. How long is the ball in the air?
Adam throws a tennis ball off a cliff, 300 metres high with an initial speed of 40 metres per second at an angle of 45° to the horizontal. Find the parametric equations that describe the position of the ball at time t.
How long is the ball in the air?
When the ball hits the ground, the vertical position y will be 0.
use the quadratic formula
The negative time value doesn't make sense so we throw it out.

10. When is the ball at its maximum height?
Adam throws a tennis ball off a cliff, 300 metres high with an initial speed of 40 metres per second at an angle of 45° to the horizontal. Find the parametric equations that describe the position of the ball at time t.
When is the ball at its maximum height?
The motion is parabolic (opening down) so maximum will be at the turning point.
What is the maximum height?

11. Determine the horizontal distance the ball traveled.
Adam throws a tennis ball off a cliff, 300 meters high with an initial speed of 40 meters per second at an angle of 45° to the horizontal. Find the parametric equations that describe the position of the ball at time t.
Determine the horizontal distance the ball traveled.
Use time in air from first part of problem.

12. Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.
Shawna has kindly given permission for this resource to be downloaded from and for it to be modified to suit the Western Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar