Parametric Functions

A parametric function describes a curve or path in the plane by expressing both x and y coordinates as separate functions of an independent variable, typically called the parameter and denoted t. Instead of writing y directly as a function of x, we write x = f(t) and y = g(t), allowing us to trace out complex curves — including ones that loop back on themselves or move in specific directions over time — that a single equation y = f(x) could never capture. The parameter t often represents time, making parametric equations a natural language for describing motion. A particle moving along a curved path has a position (x(t), y(t)) at each moment t, and the parametric equations tell us exactly where it is and, through calculus, how fast it is moving and in what direction. This connection to physics makes parametric functions essential in kinematics, engineering, computer graphics, and robotics. Parametric functions are closely tied to vectors and linear algebra. Each point (x(t), y(t)) is a position vector, and the collection of all such points as t varies traces out the parametric curve. Understanding how to eliminate the parameter, find tangent slopes, and compute arc lengths gives students powerful tools for analysing curves that appear throughout higher mathematics and applied science.

Key takeaways
  • Defining the Parameter
  • Plotting a Parametric Curve
  • Eliminating the Parameter
  • Slope of a Parametric Curve
  • Speed and Arc Length

Defining the Parameter

A parametric curve in the plane is defined by two functions: x = f(t) and y = g(t), where t ranges over some interval, such as 0 ≤ t ≤ 2π. The variable t is the parameter — it is the input that drives both coordinates simultaneously. For each value of t, we get a single point (f(t), g(t)) in the plane. As t increases, these points trace the curve in a specific direction, called the orientation of the curve.

Plotting a Parametric Curve

To plot a parametric curve, create a table of values: choose several values of t, compute x and y for each, then plot the resulting (x, y) pairs. For example, for x = cos(t) and y = sin(t) with 0 ≤ t ≤ 2π, the table gives points like (1, 0) at t = 0, (0, 1) at t = π/2, (−1, 0) at t = π, and back to (1, 0) at t = 2π. These points lie on the unit circle, traced counterclockwise. The key insight is that x² + y² = cos²(t) + sin²(t) = 1, confirming the curve is indeed the unit circle.

Eliminating the Parameter

Sometimes we want to convert a parametric curve into a single Cartesian equation. We do this by solving one equation for t and substituting into the other — or by using an algebraic identity. For x = 3t − 1 and y = 2t + 4, solve the x-equation: t = (x + 1)/3, then substitute into y: y = 2·(x+1)/3 + 4 = (2x + 2)/3 + 4 = (2x + 14)/3. This is a straight line. For trigonometric parametrisations, use Pythagorean identities like sin²(t) + cos²(t) = 1.

Slope of a Parametric Curve

The slope of the tangent line at any point on a parametric curve uses the chain rule from calculus. Since y depends on t and t depends on x, we have dy/dx = (dy/dt) ÷ (dx/dt), provided dx/dt ≠ 0. For example, if x = t² and y = t³, then dx/dt = 2t and dy/dt = 3t², so dy/dx = 3t²/(2t) = 3t/2. At t = 2, the slope is 3. This formula reveals that the slope depends on t, so the same point reached at different parameter values can have different tangents — which happens on self-intersecting curves.

Speed and Arc Length

Because t often represents time, the speed of a particle at parameter t is the magnitude of the velocity vector: speed = √((dx/dt)² + (dy/dt)²). The total arc length along the curve from t = a to t = b is L = ∫ from a to b of √((dx/dt)² + (dy/dt)²) dt. For a circle of radius r with x = r·cos(t), y = r·sin(t): dx/dt = −r·sin(t), dy/dt = r·cos(t), so the integrand is √(r²sin²(t) + r²cos²(t)) = r. Over 0 to 2π, L = 2πr, confirming the circumference formula.

The Key Equation

dydx=dydtdxdt,dxdt≠0\frac{dy}{dx} = \frac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}, \quad \frac{dx}{dt} \neq 0
  • dy/dx — Slope of tangent line (dimensionless): The instantaneous rate of change of y with respect to x along the curve — the slope of the tangent at a given point.
  • dy/dt — Rate of change of y (units of y per unit of t): How fast the y-coordinate changes as the parameter t increases; the vertical component of the velocity vector.
  • dx/dt — Rate of change of x (units of x per unit of t): How fast the x-coordinate changes as the parameter t increases; the horizontal component of the velocity vector.
  • t — Parameter (seconds (for motion) or dimensionless): The independent variable that drives both x and y; often represents time in physical applications.

Worked Example

Problem. A particle moves so that x(t) = t² − 4 and y(t) = 2t for t ≥ 0. (a) Find the Cartesian equation of the path. (b) Find the slope of the tangent at t = 3. (c) Find the speed of the particle at t = 3.

y2=4(x+4),dydx∣t=3=13,speed=210y^2 = 4(x+4),\quad \left.\frac{dy}{dx}\right|_{t=3} = \frac{1}{3},\quad \text{speed} = 2\sqrt{10}

Answer. The path is the parabola y² = 4(x + 4). At t = 3, the slope is 1/3 and the speed is 2√10.

Figure
Diagram
xyt=π/2: (0,1)t=0: (1,0)t=3π/2: (0,−1)t=π: (−1,0)x=cos t, y=sin tUnit circle: x²+y²=10 ≤ t ≤ 2π, CCW orientationr = 1
Origins · 1 minRené Descartes and how this was discovered
RD
René Descartes
1596–1650 · French

Descartes invented the Cartesian coordinate system that gives parametric functions their natural home. By associating algebraic equations with geometric curves, he unified algebra and geometry into analytic geometry. His 1637 work La Géométrie laid the foundation for expressing curves as equations, a prerequisite for the parametric approach where two separate equations jointly define a curve. Every time a student writes (x(t), y(t)) on a coordinate grid, they are working in the system Descartes created. Descartes spent much of his adult life in the Netherlands, often sleeping until noon because he did his best philosophical thinking in bed. Queen Christina of Sweden famously disrupted his habit by insisting on 5 a.m. philosophy lessons — many historians believe the harsh Stockholm winters combined with the early hours contributed to his death from pneumonia in 1650.

  • Introduction to Vectors — A parametric position (x(t), y(t)) is a vector. Understanding vector components, magnitude, and direction is essential for interpreting speed and velocity in parametric motion.
  • Linear Equations — The simplest parametric curves, x = at + b and y = ct + d, are parametric lines. Eliminating t yields a standard linear equation, connecting parametric form to the familiar y = mx + b.
  • The Unit Circle — The canonical example of a parametric curve is x = cos(t), y = sin(t), which traces the unit circle. Understanding trigonometric functions on the unit circle is central to mastering trigonometric parametrisations.
  • The Chain Rule — The slope formula dy/dx = (dy/dt)/(dx/dt) is derived using the chain rule of differentiation. The chain rule is the fundamental tool for differentiating composite functions in parametric calculus.