1. The Parameter t

A parametric equation uses a third variable, often time t, to determine x and y independently.

t = 0 t = {{ s1.t.toFixed(1) }} t = {{ s1.tMax }}
Drag time slider to trace path
x = {{ s1.x.toFixed(1) }}
y = {{ s1.y.toFixed(1) }}

2. Eliminating the Parameter

To find the direct relationship between x and y (the rectangular equation), solve for t in one equation and substitute it into the other.

Given Parametric Equations

Substitution Steps

1.
2.
3. Simplify to find

Drag chips to build equation

{{ s2.droppedM !== null ? s2.droppedM : '?' }}
{{ s2.droppedC !== null ? s2.droppedC : '?' }}
{{ chip.val }}
Equation Eliminated!

3. Circular Paths

Trigonometric functions create circular paths. Equations of the form and draw a circle of radius R.

Target Equation

Your Parametric Curve

{{ s3.a }}
{{ s3.b }}

Drag the orange handles on the graph to match the target circle.

4. Projectile Motion

A projectile's horizontal (x) and vertical (y) positions change differently over time. Gravity only affects y.

Horizontal (x)
{{ s4.vx }}
Vertical (y)
{{ s4.vy }}
{{ s4.resultMsg }}
🏆

Master of Parameters!

You've successfully controlled curves using time, eliminated parameters to find equations, built circles, and modeled projectile motion.