4.1 Parametric Functions

What are parametric functions?

A parametric function defines x and y separately in terms of a third variable, the parameter (often t): x = f(t), y = g(t). As t varies over its domain, the points (x,y) trace a curve in the plane. This lets us describe motion or curves that aren't functions of x alone, like circles.

Direction and speed of motion

Because t often represents time, parametric curves have a built-in direction of motion (orientation) as t increases. Comparing consecutive points shows whether the curve moves left/right or up/down. The rate of change of x and y with respect to t reflects how quickly the point moves along the path.

Analyzing parametric graphs

To study a parametric curve, evaluate x(t) and y(t) at several t-values to plot points, then connect them in order of increasing t. Increasing/decreasing behavior of x(t) and y(t) individually determines the shape and orientation. Some curves can be rewritten by eliminating the parameter to get an equation in x and y only.

Key terms

Parametric function

A pair of equations x = f(t), y = g(t) that define x and y in terms of a third variable, the parameter.

Parameter

The independent variable (often t) that x and y both depend on in a parametric equation.

Orientation of a parametric curve

The direction in which the curve is traced as the parameter increases.

Eliminating the parameter

Combining x(t) and y(t) algebraically to produce a single equation relating x and y directly.

Example: unit circle parametrically

x(t) = cos(t), y(t) = sin(t) for 0 ≤ t < 2π traces the unit circle x² + y² = 1.

Try an AP-style question

x = t + 1, y = t². Where is the point at t = 2?

1. (3, 2)
2. (4, 3)
3. (3, 4)
4. (2, 4)