Sketching parametric curves
Objective. Track orientation, endpoints and repeated points.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Sketching parametric curves involves plotting coordinates for increasing values of t to track the direction of travel, noting start and end points for a restricted domain and identifying any repeated points where the curve crosses itself. Example: for x=cos t, y=sin t with 0≤t≤2π, the curve traces a full circle anticlockwise starting and ending at (1,0). Arrows along the curve show the direction of increasing t, which is essential information not shown by the Cartesian equation alone.
Worked example
Sketch the parametric curve x=t, y=t³ for −2≤t≤2, describing its key features for curve sketching.
- 1Find endpoints: t=−2 gives (−2,−8); t=2 gives (2,8)
- 2Find the point at t=0: (0,0), where the curve has an inflection
- 3Note the curve increases throughout since dy/dx=3t²≥0
- 4Describe orientation: curve travels from (−2,−8) through (0,0) to (2,8)
Answer: Curve rises from (−2,−8) through (0,0) to (2,8)
Concept mastery
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Practice
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Start practisingCommon mistake. Students sketch only the Cartesian shape without marking direction arrows or endpoints, losing essential information about orientation and domain restrictions shown only in the parametric form.
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