Worksheets · Precalculus

Precalculus writing quadratic functions worksheet

In Precalculus three points with no y-intercept give a 3 by 3 system for a, b and c, solved by elimination or with a matrix. It is the same idea as fitting a curve to data.

Also taught in Algebra 1 and Algebra 2, each at its own level and with its own examples.

Select your difficulty

Pick more than one for a sheet that mixes them.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Writing Quadratic Functions

Date Period

Write the equation of each parabola.

  1. Write the equation of the parabola with vertex (2,−9) through (5,0) in vertex form
  2. Write the equation of the parabola with x-intercepts 1 and 3 through (−1,16) in factored form
  3. Write the equation of the parabola through (−3,−20),(−2,−12),(2,−20) in standard form
  4. Write the equation of the parabola in vertex form.-10-10-5-500551010
  5. Write the equation of the parabola with vertex (−4,4) through (−7,13) in vertex form
  6. Write the equation of the parabola with x-intercepts 3 and −5 through (−2,15) in factored form
  7. Write the equation of the parabola through (−4,−75),(3,2),(4,−3) in standard form
  8. Write the equation of the parabola in vertex form.-10-10-5-500551010
  9. Write the equation of the parabola with vertex (−4,−7) through (−3,−9) in vertex form
  10. Write the equation of the parabola with x-intercepts −2 and 6 through (2,16) in factored form
  11. Write the equation of the parabola through (−5,52),(−4,32),(2,38) in standard form
  12. Write the equation of the parabola in vertex form.-10-10-5-500551010
  13. Write the equation of the parabola with vertex (−1,0) through (1,−8) in vertex form
  14. Write the equation of the parabola with x-intercepts −2 and −3 through (3,60) in factored form
  15. Write the equation of the parabola through (−2,−1),(3,74),(4,107) in standard form
  16. Write the equation of the parabola in vertex form.-10-10-5-500551010
  17. Write the equation of the parabola with vertex (1,9) through (−1,13) in vertex form
  18. Write the equation of the parabola with x-intercepts −2 and 1 through (−1,2) in factored form
  19. Write the equation of the parabola through (−5,97),(−2,16),(1,−11) in standard form
  20. Write the equation of the parabola in vertex form.-10-10-5-500551010

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

Sigma Prep · sigmaprep.io/worksheets

Name

Writing Quadratic FunctionsAnswer keyVersion 1

Date Period

Write the equation of each parabola.

  1. Write the equation of the parabola with vertex (2,−9) through (5,0) in vertex formy=(x−2)2−9
  2. Write the equation of the parabola with x-intercepts 1 and 3 through (−1,16) in factored formy=2(x−1)(x−3)
  3. Write the equation of the parabola through (−3,−20),(−2,−12),(2,−20) in standard formy=−2x2−2x−8
  4. Write the equation of the parabola in vertex form.-10-10-5-500551010y=(x+4)2−1
  5. Write the equation of the parabola with vertex (−4,4) through (−7,13) in vertex formy=(x+4)2+4
  6. Write the equation of the parabola with x-intercepts 3 and −5 through (−2,15) in factored formy=−(x−3)(x+5)
  7. Write the equation of the parabola through (−4,−75),(3,2),(4,−3) in standard formy=−2x2+9x−7
  8. Write the equation of the parabola in vertex form.-10-10-5-500551010y=−2(x+3)2+6
  9. Write the equation of the parabola with vertex (−4,−7) through (−3,−9) in vertex formy=−2(x+4)2−7
  10. Write the equation of the parabola with x-intercepts −2 and 6 through (2,16) in factored formy=−(x+2)(x−6)
  11. Write the equation of the parabola through (−5,52),(−4,32),(2,38) in standard formy=3x2+7x+12
  12. Write the equation of the parabola in vertex form.-10-10-5-500551010y=−(x−4)2
  13. Write the equation of the parabola with vertex (−1,0) through (1,−8) in vertex formy=−2(x+1)2
  14. Write the equation of the parabola with x-intercepts −2 and −3 through (3,60) in factored formy=2(x+2)(x+3)
  15. Write the equation of the parabola through (−2,−1),(3,74),(4,107) in standard formy=3x2+12x+11
  16. Write the equation of the parabola in vertex form.-10-10-5-500551010y=(x+2)2+4
  17. Write the equation of the parabola with vertex (1,9) through (−1,13) in vertex formy=(x−1)2+9
  18. Write the equation of the parabola with x-intercepts −2 and 1 through (−1,2) in factored formy=−(x+2)(x−1)
  19. Write the equation of the parabola through (−5,97),(−2,16),(1,−11) in standard formy=3x2−6x−8
  20. Write the equation of the parabola in vertex form.-10-10-5-500551010y=−(x+2)2+2

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

How to do these

Write y=ax2+bx+c through (1,4), (2,9), (−1,6)

  1. The points give a + b + c = 4, 4a + 2b + c = 9 and a - b + c = 6.
  2. Subtract the first from the third: -2b = 2, so b = -1.
  3. Subtract the first from the second: 3a + b = 5, so a = 2 and c = 3.

The answer is y=2x2−x+3.

Where students go wrong

Writing -1 for (-1)². A negative x squared is positive, so the point (-1, 6) gives a - b + c = 6.

Also called equation of a parabola, quadratic from intercepts, quadratic through three points or writing quadratics.

What you can put on this worksheet

From the vertex and a point
Write the equation of the parabola with vertex (0,1) through (1,−1) in vertex form → y=−2(x)2+1
From the x-intercepts and a point
Write the equation of the parabola with x-intercepts 0 and 1 through (3,6) in factored form → y=x(x−1)
From three points
Write the equation of the parabola through (−4,17),(0,1),(2,5) in standard form → y=x2+1
Write the equation from a graph
Write the equation of the parabola in vertex form. → y=−(x−1)2+3
From a graph, opening up or down
Write the equation of the parabola. The vertex and one more point are marked. → y=12x2+1

Questions about these worksheets

Yes. Every writing quadratic functions 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 writing quadratic functions 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 5: from the vertex and a point, from the x-intercepts and a point, from three points, write the equation from a graph and from a graph, opening up or down. Tick as many as you want and set how many of each, or let it spread them evenly.

Forgetting a. Both y = (x - 2)(x + 3) and y = 2(x - 2)(x + 3) have the same intercepts; only the extra point decides which.