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.
What a learner can do afterwards
- 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 x = 3 cos t, y = 3 sin t is a circle
1 · Read
Parametric equations give x and y separately through a third variable. Think of t as time and the point as a moving dot. Pick a spread of t values, compute each pair, and join the dots. Arrows show the travel direction.
Take x equals 2t and y equals t squared. Halve x to get t, then substitute: y equals x squared over 4. The table of values traces a parabola. Direction still matters: the same shape backwards is a different parametrisation.
Matching cosine and sine spell a circle. With x as 3 cos t and y as 3 sin t, squaring and adding gives x squared plus y squared equals 9. That is a circle of radius 3 at the origin. As t grows the dot travels anticlockwise from (3, 0).
Always state what the parameter forces on x or y. The Cartesian form can describe a bigger curve than the parameter traces. Elimination is the bridge, but the bridge can widen the road.
Table the parameter, follow the arrows, and eliminate t to reach the Cartesian form.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.