Worksheets · Precalculus

Precalculus graphing ellipses worksheet

In Precalculus the ellipse comes with its foci: c² = a² - b², where a is the longer reach from the center. The larger denominator says which way the major axis runs, and the vertices and foci sit on that axis.

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

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

Sigma Prep · sigmaprep.io/worksheets

Name

Graphing Ellipses

Date Period

Graph each equation, or find what is asked.

  1. Graph x216+(y−5)24=1-10-10-5-500551010
  2. Find the vertices, co-vertices and foci of (x−2)281+(y+1)2225=1
  3. Graph x225+(y+3)29=1-10-10-5-500551010
  4. Find the vertices, co-vertices and foci of x2169+(y+4)225=1
  5. Graph (x+2)29+(y+3)225=1-10-10-5-500551010
  6. Find the vertices, co-vertices and foci of (x+1)2289+(y+2)264=1
  7. Graph (x−1)24+(y+4)216=1-10-10-5-500551010
  8. Find the vertices, co-vertices and foci of (x+2)2225+(y−2)2144=1
  9. Graph (x−4)225+(y−3)216=1-10-10-5-500551010
  10. Find the vertices, co-vertices and foci of (x−3)2169+(y−3)2144=1
  11. Graph (x+3)29+(y−5)24=1-10-10-5-500551010
  12. Find the vertices, co-vertices and foci of (x−4)2100+y264=1
  13. Graph (x+5)24+y216=1-10-10-5-500551010
  14. Find the vertices, co-vertices and foci of (x+4)225+y216=1
  15. Graph (x−1)29+y225=1-10-10-5-500551010
  16. Find the vertices, co-vertices and foci of x281+(y−1)2225=1
  17. Graph (x−3)29+(y−2)24=1-10-10-5-500551010
  18. Find the vertices, co-vertices and foci of (x+2)225+(y+4)2169=1
  19. Graph (x+6)29+(y+1)24=1-10-10-5-500551010
  20. Find the vertices, co-vertices and foci of x225+y2169=1

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

Sigma Prep · sigmaprep.io/worksheets

Name

Graphing EllipsesAnswer keyVersion 1

Date Period

Graph each equation, or find what is asked.

  1. Graph x216+(y−5)24=1-10-10-5-500551010Center (0,5), a=4, b=2
  2. Find the vertices, co-vertices and foci of (x−2)281+(y+1)2225=1Vertices (2,−16),(2,14); co-vertices (−7,−1),(11,−1); foci (2,−13),(2,11)
  3. Graph x225+(y+3)29=1-10-10-5-500551010Center (0,−3), a=5, b=3
  4. Find the vertices, co-vertices and foci of x2169+(y+4)225=1Vertices (−13,−4),(13,−4); co-vertices (0,−9),(0,1); foci (−12,−4),(12,−4)
  5. Graph (x+2)29+(y+3)225=1-10-10-5-500551010Center (−2,−3), a=3, b=5
  6. Find the vertices, co-vertices and foci of (x+1)2289+(y+2)264=1Vertices (−18,−2),(16,−2); co-vertices (−1,−10),(−1,6); foci (−16,−2),(14,−2)
  7. Graph (x−1)24+(y+4)216=1-10-10-5-500551010Center (1,−4), a=2, b=4
  8. Find the vertices, co-vertices and foci of (x+2)2225+(y−2)2144=1Vertices (−17,2),(13,2); co-vertices (−2,−10),(−2,14); foci (−11,2),(7,2)
  9. Graph (x−4)225+(y−3)216=1-10-10-5-500551010Center (4,3), a=5, b=4
  10. Find the vertices, co-vertices and foci of (x−3)2169+(y−3)2144=1Vertices (−10,3),(16,3); co-vertices (3,−9),(3,15); foci (−2,3),(8,3)
  11. Graph (x+3)29+(y−5)24=1-10-10-5-500551010Center (−3,5), a=3, b=2
  12. Find the vertices, co-vertices and foci of (x−4)2100+y264=1Vertices (−6,0),(14,0); co-vertices (4,−8),(4,8); foci (−2,0),(10,0)
  13. Graph (x+5)24+y216=1-10-10-5-500551010Center (−5,0), a=2, b=4
  14. Find the vertices, co-vertices and foci of (x+4)225+y216=1Vertices (−9,0),(1,0); co-vertices (−4,−4),(−4,4); foci (−7,0),(−1,0)
  15. Graph (x−1)29+y225=1-10-10-5-500551010Center (1,0), a=3, b=5
  16. Find the vertices, co-vertices and foci of x281+(y−1)2225=1Vertices (0,−14),(0,16); co-vertices (−9,1),(9,1); foci (0,−11),(0,13)
  17. Graph (x−3)29+(y−2)24=1-10-10-5-500551010Center (3,2), a=3, b=2
  18. Find the vertices, co-vertices and foci of (x+2)225+(y+4)2169=1Vertices (−2,−17),(−2,9); co-vertices (−7,−4),(3,−4); foci (−2,−16),(−2,8)
  19. Graph (x+6)29+(y+1)24=1-10-10-5-500551010Center (−6,−1), a=3, b=2
  20. Find the vertices, co-vertices and foci of x225+y2169=1Vertices (0,−13),(0,13); co-vertices (−5,0),(5,0); foci (0,−12),(0,12)

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

How to do these

Find the vertices and foci of (x+1)29+(y−2)225=1

  1. Center (-1, 2). The larger number, 25, is under y, so the major axis is vertical with a = 5 and b = 3.
  2. c² = 25 - 9 = 16, so c = 4.
  3. Go 5 and 4 up and down from the center.

The answer is Vertices (−1,7),(−1,−3); foci (−1,6),(−1,−2).

Where students go wrong

Adding instead of subtracting for c. For an ellipse c² = a² - b², so the foci sit inside the ellipse; c² = a² + b² is the hyperbola.

Also called plotting ellipses, ellipse graphs or major and minor axis graphs.

What you can put on this worksheet

An ellipse
Graph (x+5)29+y216=1 → Center (−5,0), a=3, b=4
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)
Foci with roots, axis lengths
Graph, and give the foci and the lengths of the axes:
x242+y260=1 → Foci (0,32) ⁣, (0,−32) major axis 415minor axis 242

Questions about these worksheets

Yes. Every graphing ellipses 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 ellipses 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: an ellipse, vertices, co-vertices and foci and foci with roots, axis lengths. Tick as many as you want and set how many of each, or let it spread them evenly.

Using 16 and 9 as the distances instead of their square roots. They are squares, so the reach is 4 and 3. The other slip is assuming the larger number always sits under the x.