Introduction
Techniques for converting parametric equations into standard rectangular form.
Past Knowledge
You can graph parametric equations and understand how traces curves.
Today's Goal
We convert parametric equations to rectangular form .
Future Success
Rectangular form reveals what curve the parametric equations trace (line, circle, parabola, etc.).
Elimination Techniques
Method 1: Direct Substitution
Solve one equation for t, substitute into the other.
Method 2: Pythagorean Identity
For trig parametrics, use .
Method 3: Algebraic Manipulation
Add, subtract, or combine equations to eliminate t.
Worked Examples
Example 1: Substitution
Eliminate t:
Solve: , substitute into y.
Solution:
Example 2: Pythagorean Identity
Eliminate t:
Use
Solution: (ellipse)
Example 3: Algebraic
Eliminate t:
Subtract:
Solution: (for x ≥ 1)
Common Pitfalls
Losing domain restrictions - State any restrictions on x or y.
Wrong identity - Use Pythagorean only for sin/cos of same angle.
Real-World Application
Path Planning in Robotics
Robots receive parametric paths but need rectangular equations to avoid obstacles.
Practice Quiz
Practice Quiz
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