Worksheets · Precalculus

Precalculus graphing conic sections worksheet

In Precalculus conics come in general form, and the first step is classifying from the squared terms: same sign and same coefficient is a circle, same sign and different coefficients an ellipse, opposite signs a hyperbola, one squared term a parabola. Completing the square then gives the center and the graph.

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

Graphing Conic Sections

Date Period

Graph each equation, or find what is asked.

  1. Graph (x+5)2+y2=16-10-10-5-500551010
  2. Graph y=−2(x+3)2+1-10-10-5-500551010
  3. Graph (x−5)216+y29=1-10-10-5-500551010
  4. Graph (x+1)29−(y+3)24=1-10-10-5-500551010
  5. Find the vertex, focus and directrix of (x−2)2=−4(y+2)
  6. Find the vertices, co-vertices and foci of (x−1)225+(y−3)29=1
  7. Find the asymptotes and foci of y236−x264=1
  8. Graph (x−3)2+(y+3)2=16-10-10-5-500551010
  9. Graph y=−(x−4)2+6-10-10-5-500551010
  10. Graph (x+3)24+(y+2)225=1-10-10-5-500551010
  11. Graph (y−3)29−(x+5)24=1-10-10-5-500551010
  12. Find the vertex, focus and directrix of (x−1)2=−8(y−3)
  13. Find the vertices, co-vertices and foci of (x+1)264+(y+2)2100=1
  14. Find the asymptotes and foci of y2576−x249=1
  15. Graph (x+2)2+(y−2)2=16-10-10-5-500551010
  16. Graph x=2(y+6)2+6-10-10-5-500551010
  17. Graph (x+5)24+y216=1-10-10-5-500551010
  18. Graph (x+2)29−(y−3)29=1-10-10-5-500551010
  19. Find the vertex, focus and directrix of (y+4)2=−16x
  20. Find the vertices, co-vertices and foci of (x+3)225+(y−1)216=1

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

Sigma Prep · sigmaprep.io/worksheets

Name

Graphing Conic SectionsAnswer keyVersion 1

Date Period

Graph each equation, or find what is asked.

  1. Graph (x+5)2+y2=16-10-10-5-500551010Center (−5,0), radius 4
  2. Graph y=−2(x+3)2+1-10-10-5-500551010Vertex (−3,1), opens down
  3. Graph (x−5)216+y29=1-10-10-5-500551010Center (5,0), a=4, b=3
  4. Graph (x+1)29−(y+3)24=1-10-10-5-500551010Center (−1,−3), opens left and right
  5. Find the vertex, focus and directrix of (x−2)2=−4(y+2)Vertex (2,−2), focus (2,−3), directrix y=−1
  6. Find the vertices, co-vertices and foci of (x−1)225+(y−3)29=1Vertices (−4,3),(6,3); co-vertices (1,0),(1,6); foci (−3,3),(5,3)
  7. Find the asymptotes and foci of y236−x264=1Asymptotes y=±34x;
    foci (0,−10),(0,10)
  8. Graph (x−3)2+(y+3)2=16-10-10-5-500551010Center (3,−3), radius 4
  9. Graph y=−(x−4)2+6-10-10-5-500551010Vertex (4,6), opens down
  10. Graph (x+3)24+(y+2)225=1-10-10-5-500551010Center (−3,−2), a=2, b=5
  11. Graph (y−3)29−(x+5)24=1-10-10-5-500551010Center (−5,3), opens up and down
  12. Find the vertex, focus and directrix of (x−1)2=−8(y−3)Vertex (1,3), focus (1,1), directrix y=5
  13. Find the vertices, co-vertices and foci of (x+1)264+(y+2)2100=1Vertices (−1,−12),(−1,8); co-vertices (−9,−2),(7,−2); foci (−1,−8),(−1,4)
  14. Find the asymptotes and foci of y2576−x249=1Asymptotes y=±247x;
    foci (0,−25),(0,25)
  15. Graph (x+2)2+(y−2)2=16-10-10-5-500551010Center (−2,2), radius 4
  16. Graph x=2(y+6)2+6-10-10-5-500551010Vertex (6,−6), opens right
  17. Graph (x+5)24+y216=1-10-10-5-500551010Center (−5,0), a=2, b=4
  18. Graph (x+2)29−(y−3)29=1-10-10-5-500551010Center (−2,3), opens left and right
  19. Find the vertex, focus and directrix of (y+4)2=−16xVertex (0,−4), focus (−4,−4), directrix x=4
  20. Find the vertices, co-vertices and foci of (x+3)225+(y−1)216=1Vertices (−8,1),(2,1); co-vertices (−3,−3),(−3,5); foci (−6,1),(0,1)

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

How to do these

Classify and graph 4x2−9y2−16x−18y−29=0

  1. The x² and y² terms have opposite signs, so it is a hyperbola.
  2. Complete the square: 4(x - 2)² - 9(y + 1)² = 36.
  3. Divide by 36: (x - 2)²/9 - (y + 1)²/4 = 1.

The answer is Hyperbola, center (2,−1), a=3, b=2.

Where students go wrong

Adding 4 and 9 when completing the square instead of 4 times 4 and 9 times 1. A number factored out in front multiplies what you add inside, so the right side changes by 16 and by -9.

Also called graphing conics, mixed conic sections or conic graphs.

What you can put on this worksheet

A circle
Graph (x−1)2+(y−3)2=9 → Center (1,3), radius 3
A parabola
Graph y=−(x−1)2+3 → Vertex (1,3), opens down
An ellipse
Graph (x+5)29+y216=1 → Center (−5,0), a=3, b=4
A hyperbola
Graph (x−3)29−(y+5)29=1 → Center (3,−5), opens left and right
Focus and directrix from the equation
Find the vertex, focus and directrix of x=112y2 → Vertex (0,0), focus (3,0), directrix x=−3
Vertices, co-vertices and foci
Find the vertices, co-vertices and foci of x2144+y2169=1 → Vertices (0,−13),(0,13); co-vertices (−12,0),(12,0); foci (0,−5),(0,5)
Asymptotes and foci
Find the asymptotes and foci of y2225−x264=1 → Asymptotes y=±158x;
foci (0,−17),(0,17)
Classify from general form and give the key points
Classify, graph, and give the key points:
4x2−9y2−24x−36y−36=0 → Hyperbolacenter (3,−2)foci (3+13,−2) ⁣, (3−13,−2)

Questions about these worksheets

Yes. Every graphing 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 graphing 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 two 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 8: a circle, a parabola, an ellipse, a hyperbola, focus and directrix from the equation, vertices, co-vertices and foci, asymptotes and foci and classify from general form and give the key points. Tick as many as you want and set how many of each, or let it spread them evenly.

Jumping to the shape before reading the signs. A minus between the two squared terms is always a hyperbola however the numbers look, and one squared term is always a parabola.