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Precalculus practice

Parametric Equations worksheets

Trace oriented parametric curves, eliminate parameters without losing domains, create parameterizations, and interpret projectile motion.

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Show parameter values, preserve every inherited restriction, and mark direction when requested.

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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

Connect a bounded parameter interval to its rectangular curve, inherited domain, and direction of motion.

Worked example

Eliminate t from x=t+1, y=t²-1, -2<=t<=2.

y=(x-1)²-1, with -1<=x<=3.

Substitute t=x-1 and translate the t interval into the corresponding x interval.

Teaching notes

Make the practice count

  • Read the parameter interval before eliminating t.
  • Use increasing t to determine orientation.
  • Keep projectile time nonnegative.

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Curve families apply to trace, eliminate, and parameterize tasks. Projectile tasks always use a quadratic constant-acceleration model. Choose the interval profile and whether students must state direction.

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Curve families

Curve families apply to trace, eliminate, and parameterize tasks. Projectile tasks always use a quadratic constant-acceleration model.

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Math SheetNameScoreTeacherDateParametric EquationsShow parameter values, preserve the inherited domain, and state the direction of motion when requested.1.Eliminate t from x = 2 cos(t), y = -2 sin(t), 0 <= t <= pi. State the inherited portion and orientation.-3-1.51.532.Make a value table and trace x = t - 1, y = 1t^2 - 1 on -2 <= t <= 2, with orientation.t-2-1012(x,y)-5-2.52.55Use exact values unless a decimal angle is requested.math-sheet.comPage 1 of 3
Math SheetNameScoreTeacherDateParametric EquationsShow parameter values, preserve the inherited domain, and state the direction of motion when requested.3.A projectile follows x = 5t and y = 5 + 14t - 4t^2 for t >= 0. Find its initial height, horizontal velocity, maximum height, and when and where it lands.4.Give a parameterization of y = 3x - 2 on -1 <= x <= 3.Use exact values unless a decimal angle is requested.math-sheet.comPage 2 of 3
Math SheetNameScoreTeacherDateParametric EquationsShow parameter values, preserve the inherited domain, and state the direction of motion when requested.5.Eliminate t from x = 6 cos(t), y = -6 sin(t), 0 <= t <= pi. State the inherited portion and orientation.-7-3.53.576.Make a value table and trace x = t - 1, y = 2t^2 - 1 on -2 <= t <= 2, with orientation.t-2-1012(x,y)-7-3.53.57Use exact values unless a decimal angle is requested.math-sheet.comPage 3 of 3

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