Introduction
Defining x and y as separate functions of a third variable, t (time).
Past Knowledge
You can graph functions and evaluate expressions.
Today's Goal
We graph parametric curves and track direction of motion.
Future Success
Parametric equations describe motion—where an object is at each moment in time.
Parametric Equations
Parametric Form
Each value of t gives a point (x, y) on the curve
Circle (radius r)
t from 0 to 2π traces full circle
Ellipse
Stretches circle by a and b
Key Concepts
- • Parameter t: Usually represents time, but can be any variable
- • Direction: As t increases, the point moves along the curve
- • Domain: The interval of t values determines which part of the curve is traced
- • Orientation: Arrows show direction of increasing t
Interactive: Watch the Circle Form
x = 2cos(t), y = 2sin(t). Watch the curve trace as t increases!
Worked Examples
Example 1: Graphing a Line
Graph the parametric equations for .
Step 1: Make a table
| t | x | y |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 2 | 2 |
| 2 | 4 | 3 |
| 3 | 6 | 4 |
Solution
Line segment from (0,1) to (6,4).
Example 2: Graphing a Parabola
Graph for .
Key Insight
Since , we have . Sideways parabola!
Direction
Starts at (4,-2), through (0,0), ends at (4,2).
Example 3: Circle Parametrization
Circle centered at (3, -1) with radius 4:
Steps
1. Start:
2. Translate: add 3 to x, -1 to y
Solution
Common Pitfalls
Ignoring the parameter domain
The range of t determines which part of the curve is traced. Always check the given interval.
Forgetting direction
Parametric curves have direction! Mark arrows to show which way t increases.
Confusing t with position
t is the parameter (often time), not a coordinate. The point's position is (x(t), y(t)).
Real-World Application
Animation and Computer Graphics
Every moving object in a video game or animation uses parametric equations! The x and y positions are functions of time t.
When you see a character walk across the screen or a ball arc through the air, parametric equations are calculating each frame's position.
Practice Quiz
Practice Quiz
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