Parametric Equations

You’ll be able to

Evaluate a parametric curve at a given parameter value
Eliminate the parameter to find a relation between x and y
Interpret the parameter as tracing a path over time

A third variable drives the motion

A parametric description gives x and y each as a function of a third variable, the parameter t (often thought of as time): x = f(t) and y = g(t). As t increases, the point (x, y) traces a path. Parametric form captures information a single y = f(x) equation cannot — the direction and timing of the motion, and even curves that fail the vertical-line test.

Eliminating the parameter

To see the underlying curve, eliminate the parameter: solve one equation for t and substitute into the other. Given x = t + 1 and y = 2t, solve the first for t = x − 1, then substitute into y = 2t to get y = 2(x − 1) = 2x − 2. The path is a straight line, though the parametric form also tells you how fast and in which direction it is traced.

Parametric form

x = f(t) · y = g(t)

Evaluate a point by plugging in a value of t. Eliminate t by solving one equation for it and substituting into the other.

Worked example

Given x = t + 1 and y = 2t, eliminate the parameter to express y in terms of x.

1.Solve the simpler equation for t: from x = t + 1, subtract 1 to get t = x − 1.
2.Substitute t = x − 1 into y = 2t: y = 2(x − 1).
3.Distribute: y = 2x − 2.

Answer: y = 2x − 2. Eliminating t turns the parametric pair into a familiar line, though the parametric form still encodes the direction of travel.

Checkpoint

Given x = t + 1 and y = 2t, eliminate the parameter to find y in terms of x.

Tip

Solve for t using whichever equation is simplest — usually the linear one. Then substitute into the other. Keeping the algebra clean here prevents sign errors in the final relation.

Checkpoint

For the parametric curve x = t², y = t + 1, what is the point when t = 2?

On the exam

After eliminating the parameter, note any restriction on the resulting curve. If t only ranges over part of an interval, the parametric curve may be just a piece of the full graph — the AP rubric checks that you respect the domain.

Answer the 2 checkpoints as you read.