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← Everything for Calculus I
Calculus IParametric and polar

Parametric curves, one moving point at a time.

Trace a curve, connect its two coordinate rules, and find its tangent. Then work through every question from the notes, with your answers checked step by step.

Points and paths · Step 1 of 6

One input gives both coordinates

x=t2{\displaystyle x} = {\displaystyle t^2}
y=t−2{\displaystyle y} = {\displaystyle t} - {\displaystyle 2}
0246810−4−3−2−101x

Arrows show increasing t. The blue dot is the current position.

(x,y)=(4.00,−4.00){\displaystyle (x,y)} = {\displaystyle (4.00,−4.00)}
A parameter is a shared input, often time. The rules x=f(t){x} = {f(t)} and y=g(t){y} = {g(t)} tell us where a point is at that moment. Use the same value of t in both rules. All the positions together form the curve.
Try it: Move t from −2 to 3. The point first goes left and up, then right and up. A curve can have two heights at the same x.
Parameter t−2.0
x=(−2.0)2=4.00{\displaystyle x} = {\displaystyle (−2.0)^2} = {\displaystyle 4.00}
Square this input for the horizontal position.
y=(−2.0)−2=−4.00{\displaystyle y} = {\displaystyle (−2.0)} - {\displaystyle 2} = {\displaystyle −4.00}
Subtract two for the vertical position.

The allowed interval is part of the question. Here the curve starts at (4, −4) and ends at (9, 1).

Check yourself
Can you use t = 1 for x and t = 2 for y to get one position?

Need a hand with parametric tangents?

Bring your practice questions. We work through them together until every type feels routine.

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