Parametric Equations
Grade 12 · Math · Study Guide
Describe curves using a third variable (parameter) and learn to find slopes and eliminate the parameter.
Parametric equations express x and y as separate functions of a third variable t (the parameter): x = f(t), y = g(t). As t varies, the point (x, y) traces a curve in the plane, with direction determined by increasing t.
To find the slope dy/dx of a parametric curve, use the chain rule: dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0. The second derivative is d²y/dx² = (d/dt[dy/dx]) / (dx/dt).
Eliminating the parameter means solving one equation for t and substituting into the other, giving a direct relationship between x and y. Not always possible analytically, but useful for recognizing standard curves like circles and ellipses.
| Quantity | Formula |
|---|---|
| Slope dy/dx | (dy/dt) / (dx/dt) |
| Second derivative d²y/dx² | (d/dt[dy/dx]) / (dx/dt) |
| Arc length | ∫ √((dx/dt)² + (dy/dt)²) dt |
| Curve | Equations |
|---|---|
| Circle (radius r) | x = r cos t, y = r sin t |
| Ellipse | x = a cos t, y = b sin t |
| Line | x = x₀ + at, y = y₀ + bt |
Key terms
- Parameter
- An independent variable t that defines both x and y as functions.
- Parametric slope
- dy/dx = (dy/dt)/(dx/dt) — slope of the tangent to a parametric curve.
- Eliminating the parameter
- Substituting to get a direct x–y equation from x(t) and y(t).
- Orientation
- The direction in which a parametric curve is traced as t increases.
Practice questions
Given x = t² and y = t³, what is dy/dx?
- 3t²/2t = 3t/2
- 2t/3t²
- t³/t²
- 3t
Show answer
dy/dx = (dy/dt)/(dx/dt) = 3t²/2t = 3t/2.
Eliminate the parameter for x = 3cos t, y = 3sin t.
Show answer
cos²t + sin²t = 1, so (x/3)² + (y/3)² = 1, giving x² + y² = 9.
At what value of t does the parametric curve x = t, y = t² have a slope of 4?
- t = 1
- t = 2
- t = 4
- t = 8
Show answer
dy/dx = 2t/1 = 2t. Set 2t = 4 → t = 2.
The orientation of a parametric curve is always clockwise.
Show answer
Orientation depends on the specific equations. For x = cos t, y = sin t with t increasing, the curve traces counterclockwise.
Find the slope of x = 2t + 1, y = t² − 3 at t = 1.
Show answer
dx/dt = 2, dy/dt = 2t. dy/dx = 2t/2 = t. At t = 1: slope = 1.
Fun facts
- Parametric equations are essential in physics — the trajectory of a projectile is naturally described as x(t) = v₀cos(θ)t and y(t) = v₀sin(θ)t − ½gt².
- Computer graphics use parametric curves (Bezier curves) to render smooth fonts and shapes in every app you use.
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