CMClearMathAcademy

Parametric Equations

A free Precalculus lesson from the “Parametric, Polar, Vectors, and Intro to Limits” unit, with a worked example and practice problems including step-by-step solutions.

Parametric equations use a third variable, often time, to describe x and y together. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Parametric, polar, vector, and limit ideas prepare students for motion, curves, and the rate-of-change thinking used in Calculus.

Worked example

Problem. For x = t + 2 and y = t^2 - 2, find the point (x, y) when t = 2.

  1. Substitute t = 2 into x = t + 2: x = 2 + 2 = 4.
  2. Substitute t = 2 into y = t^2 - 2: y = 2^2 - 2 = 4 - 2 = 2.
  3. So the point is (4, 2).

Answer: (4, 2)

Practice problems

1. For x = t + 2 and y = t^2 - 2, find the point (x, y) when t = 2.

Show solution
  1. Substitute t = 2 into x = t + 2: x = 2 + 2 = 4.
  2. Substitute t = 2 into y = t^2 - 2: y = 2^2 - 2 = 4 - 2 = 2.
  3. So the point is (4, 2).

Answer: (4, 2)

2. For x = 4t + 3 and y = t^2 + 3, find the point (x, y) when t = 3.

Show solution
  1. Substitute t = 3 into x = 4t + 3: x = 4(3) + 3 = 12 + 3 = 15.
  2. Substitute t = 3 into y = t^2 + 3: y = 3^2 + 3 = 9 + 3 = 12.
  3. So the point is (15, 12).

Answer: (15, 12)

3. For x = 2t + 4 and y = t^2 + 4, find only the x-coordinate when t = 4.

Show solution
  1. Core Practice: First identify exactly what the question is asking: For x = 2t + 4 and y = t^2 + 4, find only the x-coordinate when t = 4.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use the x-rule x = 2t + 4.
  4. Substitute t = 4: x = 2(4) + 4 = 8 + 4.
  5. So x = 12.
  6. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

4. For x = 3t + 1 and y = t^2 + 5, find only the y-coordinate when t = 1.

Show solution
  1. Core Practice: First identify exactly what the question is asking: For x = 3t + 1 and y = t^2 + 5, find only the y-coordinate when t = 1.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use the y-rule y = t^2 + 5.
  4. Substitute t = 1: y = 1^2 + 5 = 1 + 5.
  5. So y = 6.
  6. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

Answer: 6

5. A point moves with x = 4t + 2 and y = t^2 + 1. Where is it at the start, t = 0?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A point moves with x = 4t + 2 and y = t^2 + 1. Where is it at the start, t = 0?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. At t = 0, x = 4(0) + 2 = 2.
  4. At t = 0, y = 0^2 + 1 = 1.
  5. So the starting point is (2, 1).
  6. Check the result by substituting or estimating: the response should match (2, 1) and make sense in the original problem.

Answer: (2, 1)

Practice this interactively with instant feedback and an AI tutor.

Practice Parametric Equations Take the free placement check

More Precalculus lessons