Worksheets · Precalculus

Parametric equations worksheet

Two equations, one parameter, one curve. Solve whichever equation is easier for t and put it into the other, and the t disappears leaving an equation in x and y.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Parametric Equations

Date Period

Eliminate the parameter, sketch the curve, or write parametric equations.

  1. Eliminate the parameter

    Eliminate the parameter: x=t−6,y=4t2+8
  2. Eliminate the parameter: x=et, y=e2t−1
  3. Eliminate the parameter:
    x=2t−1,y=−2t+2
  4. Eliminate the parameter: x=2sec⁡t, y=3tan⁡t
  5. Eliminate the parameter:
    x=t+1,y=−2t2−8
  6. Eliminate the parameter: x=t, y=t+6
  7. Eliminate the parameter: x=t+6,y=−t2+6
  8. Eliminate the parameter: x=5sec⁡t, y=tan⁡t
  9. Eliminate the parameter: x=t−7,y=4t+5
  10. Eliminate the parameter: x=tan⁡t, y=sec⁡t

  1. Sketch the curve and show its direction

    Sketch x=t+1, y=t2+1 for −2≤t≤2 and show its direction.−10−10−5−5551010
  2. Sketch x=−t−1, y=−t−3 for −3≤t≤3 and show its direction.−10−10−5−5551010
  3. Sketch x=t2+3, y=t−3 for −2≤t≤2 and show its direction.−10−10−5−5551010
  4. Sketch x=−t+3, y=t2−1 for −2≤t≤2 and show its direction.−10−10−5−5551010
  5. Sketch x=t−2, y=t−3 for −3≤t≤3 and show its direction.−10−10−5−5551010

  1. Write parametric equations for a segment

    Write parametric equations for the segment from (6,−1) to (0,−5).
  2. Write parametric equations for the segment from (4,6) to (−3,3).
  3. Write parametric equations for the segment from (6,3) to (−1,−3).
  4. Write parametric equations for the segment from (3,−4) to (2,6).
  5. Write parametric equations for the segment from (−3,−5) to (2,1).

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Parametric EquationsAnswer keyVersion 1

Date Period

Eliminate the parameter, sketch the curve, or write parametric equations.

  1. Eliminate the parameter

    Eliminate the parameter: x=t−6,y=4t2+8y=4x2+48x+152
  2. Eliminate the parameter: x=et, y=e2t−1y=x2−1, x>0
  3. Eliminate the parameter:
    x=2t−1,y=−2t+2    y=−x+1
  4. Eliminate the parameter: x=2sec⁡t, y=3tan⁡tx24−y29=1
  5. Eliminate the parameter:
    x=t+1,y=−2t2−8    y=−2x2+4x−10
  6. Eliminate the parameter: x=t, y=t+6y=x2+6, x≥0
  7. Eliminate the parameter: x=t+6,y=−t2+6y=−x2+12x−30
  8. Eliminate the parameter: x=5sec⁡t, y=tan⁡tx225−y2=1
  9. Eliminate the parameter: x=t−7,y=4t+5y=4x+33
  10. Eliminate the parameter: x=tan⁡t, y=sec⁡ty2−x2=1

  1. Sketch the curve and show its direction

    Sketch x=t+1, y=t2+1 for −2≤t≤2 and show its direction.−10−10−5−551010From (−1,5) through (1,1) to (3,5)
  2. Sketch x=−t−1, y=−t−3 for −3≤t≤3 and show its direction.−10−10−5−5551010From (2,0) through (−1,−3) to (−4,−6)
  3. Sketch x=t2+3, y=t−3 for −2≤t≤2 and show its direction.−10−10−5−5551010From (7,−5) through (3,−3) to (7,−1)
  4. Sketch x=−t+3, y=t2−1 for −2≤t≤2 and show its direction.−10−10−5−5551010From (5,3) through (3,−1) to (1,3)
  5. Sketch x=t−2, y=t−3 for −3≤t≤3 and show its direction.−10−10−5−5551010From (−5,−6) through (−2,−3) to (1,0)

  1. Write parametric equations for a segment

    Write parametric equations for the segment from (6,−1) to (0,−5).Sample answer: x=6−6t, y=−1−4t, 0≤t≤1
  2. Write parametric equations for the segment from (4,6) to (−3,3).Sample answer: x=4−7t, y=6−3t, 0≤t≤1
  3. Write parametric equations for the segment from (6,3) to (−1,−3).Sample answer: x=6−7t, y=3−6t, 0≤t≤1
  4. Write parametric equations for the segment from (3,−4) to (2,6).Sample answer: x=3−t, y=−4+10t, 0≤t≤1
  5. Write parametric equations for the segment from (−3,−5) to (2,1).Sample answer: x=−3+5t, y=−5+6t, 0≤t≤1

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

x=t+2,y=3t−1

  1. From the first, t is x minus 2.
  2. Put that into the second.
  3. 3(x - 2) - 1 is 3x - 7.

The answer is y=3x−7.

Where students go wrong

Reaching for substitution when the curve is trigonometric. With x and y built from cos t and sin t you do not solve for t at all: square both and add, and the Pythagorean identity takes the parameter out for you.

Also called eliminating the parameter, parametric to rectangular or parametrised curves.

What you can put on this worksheet

Eliminate the parameter
Eliminate the parameter: x=t+1,y=−3t−4 → y=−3x−1
Sketch the curve and its direction
Sketch x=t, y=t2+1 for −2≤t≤2 and show its direction. → From (−2,5) through (0,1) to (2,5)
Parametric equations for a segment
Write parametric equations for the segment from (0,1) to (3,−5). → Sample answer: x=3t, y=1−6t, 0≤t≤1
Eliminate the parameter: more forms
Eliminate the parameter: x=et, y=3e−t → y=3x, x>0
Parametrize with a given parameter
Write parametric equations for y=2x2+3x−5 using the parameter t=x−3. → x=t+3, y=2t2+15t+22
Faster, or the other way
The curve y=2x2+3 is traced by x=t, y=2t2+3. Write equations that trace it twice as fast, in the opposite direction. → x=−2t, y=8t2+3
Sketch a trig curve with its direction
Sketch x=2cos⁡t, y=−4sin⁡t for 0≤t≤2π, showing the direction. → Ellipse x24+y216=1, starting at (2,0), going clockwise

Questions about these worksheets

Yes. Every parametric equations sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the parametric equations questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 7: eliminate the parameter, sketch the curve and its direction, parametric equations for a segment, eliminate the parameter: more forms, parametrize with a given parameter, faster, or the other way and sketch a trig curve with its direction. Tick as many as you want and set how many of each, or let it spread them evenly.

Reaching for substitution when the curve is trigonometric. With x and y built from cos t and sin t you do not solve for t at all: square both and add, and the Pythagorean identity takes the parameter out for you.