--- title: "Parametric equations of curves" description: "Describe a curve by giving x and y separately in terms of a parameter, plot it from a table of parameter values, and convert to Cartesian form by eliminating the parameter." canonical: https://lightmysky.com/learn/mathematics/parametric-equations-of-curves-mt_7NC7SCeU2P source: https://lightmysky.com/learn/mathematics/parametric-equations-of-curves-mt_7NC7SCeU2P.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Parametric equations of curves Describe a curve by giving x and y separately in terms of a parameter, plot it from a table of parameter values, and convert to Cartesian form by eliminating the parameter. Subject: Mathematics · Area: Geometry · Ages 16 to 17 Page: https://lightmysky.com/learn/mathematics/parametric-equations-of-curves-mt_7NC7SCeU2P ## Ready when they can - Plot the curve x = 2t, y = t² for a range of t values - Eliminate t to get the Cartesian equation of that curve - Use sin²t + cos²t = 1 to show that x = 3 cos t, y = 3 sin t traces a circle for 0 <= t <= 2pi radians ## Lesson: Curves drawn by a moving dot A parameter is an input used to calculate both coordinates. It may represent time or an angle. For x = 2t and y = t², use the same allowed t in both equations to get one point (x, y). Make a table with columns for t, x and y. Plot each pair with x horizontal and y vertical. For these smooth curves, sketch a smooth path through the points in increasing t order. Use each t in both equations, then plot each (x, y) on standard axes. From the origin, move horizontally by x, then vertically by y. Positive means right or up; negative means left or down. Here (-4, 4) is 4 units left and 4 up, while (4, 4) is 4 right and 4 up. The points lie on an upward U whose lowest point is (0, 0). Sketch a smooth curve through them. As t increases, add arrows down the left branch to the origin, then up the right branch. **Example.** Eliminating t gives a relation between x and y without t. For x = 2t and y = t², solve t = x/2, then substitute: y = (x/2)² = x²/4. With x = t + 1 and y = t², use t = x - 1 to get y = (x - 1)². Expand by multiplying each term in one bracket by each term in the other: x² - x - x + 1 = x² - 2x + 1. To find y at a known x, solve for t first, then use that same t in the y equation. Keep the parameter range when eliminating t. For x = 2t, y = t² and -1 <= t <= 3, the endpoints give x = -2 and x = 6. Since x increases with t, every x between them is reached. The Cartesian description is y = x²/4 with -2 <= x <= 6. Without this restriction, the equation includes points outside the traced part of the parabola. For x = r cos t and y = r sin t, where r > 0, use cos²t + sin²t = 1: x² + y² = r². This is a circle of radius r centred at (0, 0). Angles here are in radians; pi/2 is a quarter turn, or 90 degrees. At t = 0, cos t = 1 and sin t = 0. At t = pi/2, cos t = 0 and sin t = 1. For 0 <= t <= 2pi, all angles of one turn are covered, so the whole circle is traced once. *(drawing: For radius 3, these positions occur in this order. The last position matches the first after one full turn.)* Arrows on a sketch show travel as t increases. On standard axes, x = r cos t and y = r sin t, with r > 0, move anticlockwise from (r, 0) when t starts at 0. Changing y to -r sin t gives (r, 0) at t = 0 and (0, -r) at t = pi/2, so travel is clockwise. For 0 <= t <= 2pi, both versions trace the same set of points. They are different parametrisations of the same circle. An angle range can restrict a circle to an arc. For x = r cos t and y = r sin t, with r > 0 and 0 <= t <= pi, sine is nonnegative, so y >= 0. The point travels from (r, 0) through (0, r) to (-r, 0). It traces the whole upper semicircle, including both endpoints. Its Cartesian description is x² + y² = r² with y >= 0. **Recap.** Use the same t for both coordinates. Plot in increasing t order, eliminate t, and keep the restrictions from its allowed range. *The lesson and questions were written for this dataset. OpenStax and Khan Academy records informed the mathematical checks and teaching design; their prose is not reproduced.* ## Practice 14 questions on this page, each with its working shown. ## Optional connections - [The Pythagorean and Quotient Identities](https://lightmysky.com/learn/mathematics/the-pythagorean-and-quotient-identities-mt_NXBq37vxQh) ## Opens up - [Conic Sections: Ellipse, Parabola and Hyperbola](https://lightmysky.com/learn/mathematics/conic-sections-ellipse-parabola-and-hyperbola-mt_V4YSI3a_Eu) - [Vector-Valued Functions and Motion Along a Curve](https://lightmysky.com/learn/mathematics/vector-valued-functions-and-motion-along-a-curve-mt_Xcjt6exMy3)