Cogito
AP Precalculus · Chapter 8 · Lesson 1
Parametric Functions
A third variable driving both coordinates.
12 problems · about 21 minutes · AP Precalculus 4.1, 4.2
What this lesson teaches
The student interprets parametric functions and eliminates the parameter.
- Parametric functions give x and y as functions of a parameter t.
- Eliminating t recovers the rectangular equation of the path.
- Only the parametric form carries direction and timing.
Warm Up
Straightforward practice. Get the method working first.
5 problemsx = 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.
Build It Up
The same ideas with more to keep track of.
3 problemsx = 4t, y = 8t. Eliminating t gives y = kx. What is k?
Answer 2
Why t = x ÷ 4.
x = −t, y = 3t. As t increases, does x rise or fall? 1 rise, 2 fall.
Answer 2
Why The negative reverses it.
x = t, y = t². Eliminating t gives y = xⁿ. What is n?
Answer 2
Why Substitute directly.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsWhat 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 Circle: x = cos t and y = sin t. What is x² + y²?
Answer 1
Why 1.