Worksheets · Precalculus

Classifying conic sections worksheet for Precalculus

Precalculus goes past naming the conic: once the squared terms say which it is, you complete the square to reach standard form and read off the center or vertex. Half the questions only ask for the name; the rest name the conic and then write it in standard form, starting from a general form with every term present.

Also taught in Algebra 2, at its own level and with its own examples.

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

Classifying Conic Sections

Date Period

Name each conic, and write it in standard form when asked.

  1. Classify the conic section

    3x2+7y2−10x+30y+17=0
  2. 6x2−y2+5x+13y+2=0
  3. 4x2+34x−16y−2=0
  4. x2+y2+3x+18y−24=0
  5. 6x2+6y2+34x+22y+2=0
  6. 6x2−x+11y−28=0
  7. 2x2+7y2+7x−12y+1=0
  8. x2+17x+9y−25=0
  9. 2x2−2y2−8x+16y+18=0
  10. x2−11x+9y−30=0

  1. Classify, then write in standard form

    Classify, then write in standard form:
    9x2+25y2−54x+150y+81=0
  2. Classify, then write in standard form:
    16x2+25y2+96x+200y+144=0
  3. Classify, then write in standard form:
    x2+9y2+8x−72y+151=0
  4. Classify, then write in standard form: x2−8x+12y−20=0
  5. Classify, then write in standard form: x2−12y−24=0
  6. Classify, then write in standard form: x2+2x−8y−23=0
  7. Classify, then write in standard form:
    9x2+25y2−36x+50y−164=0
  8. Classify, then write in standard form: x2−6x+4y+5=0
  9. Classify, then write in standard form:
    9x2−25y2−54x−150y−369=0
  10. Classify, then write in standard form:
    16x2+25y2−64x−100y−236=0

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

Sigma Prep · sigmaprep.io/worksheets

Name

Classifying Conic SectionsAnswer keyVersion 2

Date Period

Name each conic, and write it in standard form when asked.

  1. Classify the conic section

    3x2+7y2−10x+30y+17=0Ellipse
  2. 6x2−y2+5x+13y+2=0Hyperbola
  3. 4x2+34x−16y−2=0Parabola
  4. x2+y2+3x+18y−24=0Circle
  5. 6x2+6y2+34x+22y+2=0Circle
  6. 6x2−x+11y−28=0Parabola
  7. 2x2+7y2+7x−12y+1=0Ellipse
  8. x2+17x+9y−25=0Parabola
  9. 2x2−2y2−8x+16y+18=0Hyperbola
  10. x2−11x+9y−30=0Parabola

  1. Classify, then write in standard form

    Classify, then write in standard form:
    9x2+25y2−54x+150y+81=0    ellipse, (x−3)225+(y+3)29=1
  2. Classify, then write in standard form:
    16x2+25y2+96x+200y+144=0    ellipse, (x+3)225+(y+4)216=1
  3. Classify, then write in standard form:
    x2+9y2+8x−72y+151=0    ellipse, (x+4)29+(y−4)2=1
  4. Classify, then write in standard form: x2−8x+12y−20=0parabola, (x−4)2=−12(y−3)
  5. Classify, then write in standard form: x2−12y−24=0parabola, x2=12(y+2)
  6. Classify, then write in standard form: x2+2x−8y−23=0parabola, (x+1)2=8(y+3)
  7. Classify, then write in standard form:
    9x2+25y2−36x+50y−164=0    ellipse, (x−2)225+(y+1)29=1
  8. Classify, then write in standard form: x2−6x+4y+5=0parabola, (x−3)2=−4(y−1)
  9. Classify, then write in standard form:
    9x2−25y2−54x−150y−369=0    hyperbola, (x−3)225−(y+3)29=1
  10. Classify, then write in standard form:
    16x2+25y2−64x−100y−236=0    ellipse, (x−2)225+(y−2)216=1

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

How to do these

Classify, then write in standard form:
4x2+y2−16x+6y+9=0

  1. Both squared terms are positive with different coefficients, so it is an ellipse.
  2. Group and complete the square: 4(x - 2)² + (y + 3)² = -9 + 16 + 9 = 16.
  3. Divide by 16.

The answer is ellipse, (x−2)24+(y+3)216=1.

Where students go wrong

Calling it a circle whenever both squared terms are positive. A circle needs equal coefficients on x² and y²; 4x² + y² is an ellipse, and the two different denominators in standard form show it.

Also called identifying conics or conic sections.

What you can put on this worksheet

Name the conic from its equation
5x2−4y2+5x−4y−4=0 → Hyperbola
Classify, then write in standard form
Classify, then write in standard form: x2−2x+8y−15=0 → parabola, (x−1)2=−8(y−2)
Standard forms, sideways parabolas, negative leads
(x−1)24−(y+6)249=1 → Hyperbola

Questions about these worksheets

Yes. Every classifying conic sections 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 classifying conic sections 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 3: name the conic from its equation, classify, then write in standard form and standard forms, sideways parabolas, negative leads. Tick as many as you want and set how many of each, or let it spread them evenly.

Looking at the linear terms. The x and y terms move the curve around but never change what it is: only the squared coefficients decide, and if one of them is missing the answer is a parabola whatever else is there.