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BetterGrades Precalculus · Unit 14 · Unit map

Parametric, Polar, and Complex Representations

For x=2t-1 and y=t^2, -2≤t≤3, find endpoints, direction, and the Cartesian curve.

Precalculus · Unit 14

The lesson path

For x=2t-1 and y=t^2, -2≤t≤3, find endpoints, direction, and the Cartesian curve.

Core sequence

Parametric, Polar, and Complex Representations

Move in order when the material is new, or enter at the exact skill you need.

  1. 01Parametric equations and orientationInterpret x=f(t),y=g(t)x=f(t), y=g(t) as a directed path with a parameter interval.
  2. 02Eliminating a parameterEliminate parameters while preserving interval restrictions and lost orientation information.
  3. 03Parametric motion and vector-valued positionUse r(t)=<x(t),y(t)>r(t)=<x(t),y(t)> to describe position, displacement, and average velocity.
  4. 04Intersections, repeated points, and multiple parameter valuesDistinguish curve intersections from self-intersections and repeated positions.
  5. 05Polar coordinates and nonuniquenessRepresent points as (r,theta), use negative radius, and generate equivalent polar coordinates.
  6. 06Cartesian and polar conversionConvert points and equations using x=rx=r cos theta, y=ry=r sin theta, and r2=x2+y2r^2=x^2+y^2.
  7. 07Graphing polar equationsTrace r=f(theta)r=f(theta) using tables, signed radius, and interval selection.
  8. 08Polar symmetry and repeated tracingTest symmetry and choose intervals that trace a polar curve exactly once.
  9. 09Common polar families and polar conicsAnalyze parameter effects in circles, cardioids, limacons, roses, lemniscates, spirals, and conics.
  10. 10Complex numbers in polar formRepresent complex numbers by modulus and argument and convert between rectangular and polar forms.
  11. 11Complex multiplication, division, and powersMultiply moduli, add arguments, divide moduli, subtract arguments, and apply De Moivre's theorem.
  12. 12Roots of complex numbersFind all nth roots of a complex number and interpret their equal angular spacing.
Source record

References used for this unit

  • Lippman & Rasmussen, Precalculus Vol. 2, 8.2, 8.3, 8.6, 9.4
  • Sundstrom & Schlicker, Trigonometry, Chapter 5
  • Yoshiwara, Trigonometry, Chapter 10
  • Corral, Trigonometry, 6.3-6.4

The learner copy is original BetterGrades material informed by these rights-separated references; no long source passage is reproduced.