Parametric Equations

Let a third variable drive both coordinates.

12 problems · about 22 minutes · TEKS P.3.C, TEKS P.3.E

What this lesson teaches

The student writes and interprets parametric equations, and eliminates the parameter to recover a rectangular equation.

Parametric equations give x and y as functions of a parameter t.
Eliminating t recovers the rectangular equation of the path.
The parametric form also carries direction and timing, which the rectangular form forgets.

Straightforward practice. Get the method working first.

5 problems

x = t − 1, y = 3t. What is y when t = 5?
Answer 15
Why 15.
What is lost when the parameter is eliminated?
Answer The direction of travel and any restriction on t.
Why The motion information.
x = 2t, y = t². What is x when t = 3?
Answer 6
Why 2 × 3.
Same equations. What is y when t = 3?
Answer 9
Why 3².
x = t + 5, y = t. What is y when x = 9?
Answer 4
Why t = x − 5.

The same ideas with more to keep track of.

3 problems

x = cos t, y = sin t. What is x² + y²?
Answer 1
Why The Pythagorean identity.
x = 4t, y = 8t. Eliminating t gives y = kx. What is k?
Answer 2
Why t = x ÷ 4, so y = 2x.
x = t, y = t². Eliminating t gives y = xⁿ. What is n?
Answer 2
Why Substitute directly.

Mixed problems. Work out what kind of question it is before you start.

4 problems

What does a parametric form carry that a rectangular one does not?
Answer The direction of travel; When the object reaches each point
Why Both forms give the shape; only one gives the motion.
x = 5t, y = 3t. What is y when t = 2?
Answer 6
Why 3 × 2.
The Position: x = 3t and y = t + 2. What is x when t = 4?
Answer 12
Why 12.
The Other Coordinate: Same equations. What is y when t = 4?
Answer 6
Why 6.