Skill: parametric-equations

Parametric Equations

Describe motion with parametric equations: plot points from x(t) and y(t), eliminate the parameter, and track direction along the curve.

1 What are parametric equations?

Instead of y as a function of x, both x and y are functions of a third variable, the parameter t — often thought of as time.

Parametric form

x = f(t), y = g(t), for t in an interval

Each t gives one point (x(t), y(t)). As t increases, the point moves along the curve — parametrics describe the journey, not just the path. The t-interval restricts which part of the curve is drawn: always check the endpoints.

Plot x = 2t + 1, y = t − 3 for t = 0, 1, 2

t = 0 → (1, −3); t = 1 → (3, −2); t = 2 → (5, −1). Plot the three points and connect them — the motion goes up and to the right as t increases.

2 Eliminating the parameter

Solve one equation for t, substitute into the other — the parameter disappears, leaving a rectangular (x-y) equation.

Linear elimination

x = at + b, y = ct + d → t = x − ba → y = ca(x − b) + d

The rectangular form shows the path but loses the direction and the t-interval — always report those separately. For circles, x = h + r cos t, y = k + r sin t eliminates to (x − h)² + (y − k)² = r² via cos²t + sin²t = 1.

Eliminate x = 2t + 1, y = 4t + 3

t = (x − 1)/2, so y = 4·(x − 1)/2 + 3 = 2(x − 1) + 3 = y = 2x + 1.

3 Worked examples

Plot, eliminate, and track direction — each claim verified independently.

Example 1 Evaluate at t

For x = 2t + 1, y = 3t − 2, find the point when t = 4.

x = 2(4) + 1 = 9, y = 3(4) − 2 = 10: (9, 10).

Example 2 Evaluate at t

For x = 5, y = −t + 3, find the point when t = 2.

x = 5 (constant!), y = −2 + 3 = 1: (5, 1).

Example 3 Circular motion

For x = 5 cos t, y = 5 sin t, find the point when t = 90°.

x = 5 cos 90° = 0, y = 5 sin 90° = 5: (0, 5).

Example 4 Eliminate the parameter

Eliminate t from x = 2t + 1, y = 4t + 3.

t = (x − 1)/2 → y = 2(x − 1) + 3 = y = 2x + 1.

Example 5 Direction of motion

For x = 2t, y = −3t + 1, as t increases does x move left or right?

a = 2 > 0, so x increases with t: the point moves right (and down, since c = −3).

Example 6 Endpoints of the interval

For x = 2t + 1, y = 4t + 3 with 0 ≤ t ≤ 2, find the starting point.

At t = 0: (2·0 + 1, 4·0 + 3) = (1, 3). The curve is only the segment from (1, 3) to (5, 11) — not the whole line.

Example 7 Shifted circle

For x = 2 + 4 cos t, y = −1 + 4 sin t, find the point when t = 180°.

x = 2 + 4(−1) = −2, y = −1 + 0 = −1: (−2, −1).

4 Common mistakes

Two ways parametric problems go wrong.

Ignoring the parameter interval

Wrong: graphing x = 2t + 1, y = 4t + 3 for 0 ≤ t ≤ 2 as the entire line y = 2x + 1.
Right: the interval restricts the curve to the segment from (1, 3) to (5, 11). Always compute the endpoints.

Eliminating the parameter but losing the direction

Wrong: reporting only y = 2x + 1 and stopping.
Right: the rectangular form shows the path; the parametrics show the journey — report the direction (here, right and up as t increases) too.

5 Key vocabulary

Words to know

Parameter (t) — the independent variable driving both x and y.
Eliminate the parameter — solve for t and substitute to get a rectangular equation.
Rectangular equation — the x-y equation of the path (direction and interval lost).
Orientation / direction — which way the point travels as t increases.

6 Quick check

Try these before moving on — click to reveal each answer.

For x = 3t − 1, y = t + 2, find the point when t = 5.

x = 15 − 1 = 14, y = 5 + 2 = 7: (14, 7).

Eliminate t: x = t + 4, y = 3t − 2.

t = x − 4 → y = 3(x − 4) − 2 = y = 3x − 14.

x = 1 + 2 cos t, y = 3 + 2 sin t traces which circle?

Center (1, 3), radius 2: (x − 1)² + (y − 3)² = 4.

7 Next steps

Now drill the skill with endless randomized problems.

Practice Try the Tool Explore the Lab Print a Worksheet Play the Game