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Precalculus practice

Polar Function Graphs worksheets

Build key-value tables, test symmetry, graph full polar curves, and identify circle, cardioid, limacon, rose, and lemniscate families.

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Use signed radii, plot in increasing-angle order, and complete one full trace.

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Coordinate descriptions carry restrictions

These sheets connect parametric, rectangular, polar, and vector representations while keeping the interval, orientation, quadrant, or algebraic restriction that makes each conversion equivalent.

Learning objective

What this activity builds

Trace standard polar families once over a correct full interval without revealing graph-task answers on the student sheet.

Worked example

What family is r=2(1+cos(theta))?

A right-facing cardioid.

The radius reaches 4 at theta=0 and reaches the pole at theta=pi.

Teaching notes

Make the practice count

  • Plot negative r on the opposite ray.
  • Count rose petals before choosing a tracing interval.
  • Test symmetry algebraically before relying on appearance.

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Choose graph, identify, or mixed prompts across bounded standard polar families.

Worksheet settings

Curve families
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Math SheetNameScoreTeacherDatePolar Function GraphsUse the stated full-trace interval, test the requested symmetry, and plot the key-angle table before drawing the curve.1.Graph r^2 = 4 cos(2theta) on 0 <= theta <= 2pi. Test polar-axis symmetry with theta -> -theta, vertical-axis symmetry with theta -> pi - theta, andpole symmetry with theta -> theta + pi. Complete the key-value table; for a squared-radius equation, tabulate the nonnegative root.theta0pi/4pi/2pi3pi/22pir123numbered radial scale2.Identify the completed polar graph's family and write a matching equation. Use the numbered radial scale. State its symmetries.12345completed graph with numbered radial scaleUse exact values unless a decimal angle is requested.math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDatePolar Function GraphsUse the stated full-trace interval, test the requested symmetry, and plot the key-angle table before drawing the curve.3.Graph r = 4(1 + cos(theta)) on 0 <= theta <= 2pi. Test polar-axis symmetry with theta -> -theta, vertical-axis symmetry with theta -> pi - theta, andpole symmetry with theta -> theta + pi. Complete the key-value table; for a squared-radius equation, tabulate the nonnegative root.theta0pi/4pi/2pi3pi/22pir123456789numbered radial scale4.Identify the completed polar graph's family and write a matching equation. Use the numbered radial scale. State its symmetries.123456completed graph with numbered radial scaleUse exact values unless a decimal angle is requested.math-sheet.comPage 2 of 2

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