Worksheets · Precalculus

Vertex of a parabola worksheet for Precalculus

Precalculus keeps the vertex but adds the focus and directrix, so the parabola becomes a conic defined by distance. Write the equation as y - k = (1/(4p))(x - h)², or the sideways version with x and y swapped, then read the vertex, the value of p, and the direction it opens.

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

Vertex of a Parabola

Date Period

Answer each question.

  1. Find the vertex of y=ax2+bx+cy = ax^2 + bx + c

    Find the vertex of y=6x2+60x+152
  2. Find the vertex of y=6x2−48x+89
  3. Find the vertex of y=6x2+12x−2
  4. Find the vertex of y=7x2−14x−17
  5. Find the vertex of y=−8x2−48x−76
  6. Find the vertex of y=−6x2+120x−595
  7. Find the vertex of y=7x2−24

  1. Write a parabola from focus and directrix

    Write the equation of the parabola with focus (5,0) and directrix y=4
  2. Write the equation of the parabola with focus (7,−6) and directrix x=5
  3. Write the equation of the parabola with focus (−6,−1) and directrix y=1
  4. Write the equation of the parabola with focus (−2,2) and directrix y=−4
  5. Write the equation of the parabola with focus (−4,4) and directrix x=2
  6. Write the equation of the parabola with focus (0,−5) and directrix y=−3
  7. Write the equation of the parabola with focus (−7,−1) and directrix x=−3

  1. Find the vertex, focus and directrix

    Find the vertex, focus and directrix of y=−116(x+1)2+3
  2. Find the vertex, focus and directrix of x=−14(y−1)2
  3. Find the vertex, focus and directrix of x=14(y−4)2−4
  4. Find the vertex, focus and directrix of x=−14(y−2)2+1
  5. Find the vertex, focus and directrix of y=116(x−1)2
  6. Find the vertex, focus and directrix of y=−116(x−3)2−2

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

Sigma Prep · sigmaprep.io/worksheets

Name

Vertex of a ParabolaAnswer keyVersion 2

Date Period

Answer each question.

  1. Find the vertex of y=ax2+bx+cy = ax^2 + bx + c

    Find the vertex of y=6x2+60x+152(−5,2)
  2. Find the vertex of y=6x2−48x+89(4,−7)
  3. Find the vertex of y=6x2+12x−2(−1,−8)
  4. Find the vertex of y=7x2−14x−17(1,−24)
  5. Find the vertex of y=−8x2−48x−76(−3,−4)
  6. Find the vertex of y=−6x2+120x−595(10,5)
  7. Find the vertex of y=7x2−24(0,−24)

  1. Write a parabola from focus and directrix

    Write the equation of the parabola with focus (5,0) and directrix y=4y=−18(x−5)2+2
  2. Write the equation of the parabola with focus (7,−6) and directrix x=5x=14(y+6)2+6
  3. Write the equation of the parabola with focus (−6,−1) and directrix y=1y=−14(x+6)2
  4. Write the equation of the parabola with focus (−2,2) and directrix y=−4y=112(x+2)2−1
  5. Write the equation of the parabola with focus (−4,4) and directrix x=2x=−112(y−4)2−1
  6. Write the equation of the parabola with focus (0,−5) and directrix y=−3y=−14x2−4
  7. Write the equation of the parabola with focus (−7,−1) and directrix x=−3x=−18(y+1)2−5

  1. Find the vertex, focus and directrix

    Find the vertex, focus and directrix of y=−116(x+1)2+3Vertex (−1,3), focus (−1,−1), directrix y=7
  2. Find the vertex, focus and directrix of x=−14(y−1)2Vertex (0,1), focus (−1,1), directrix x=1
  3. Find the vertex, focus and directrix of x=14(y−4)2−4Vertex (−4,4), focus (−3,4), directrix x=−5
  4. Find the vertex, focus and directrix of x=−14(y−2)2+1Vertex (1,2), focus (0,2), directrix x=2
  5. Find the vertex, focus and directrix of y=116(x−1)2Vertex (1,0), focus (1,4), directrix y=−4
  6. Find the vertex, focus and directrix of y=−116(x−3)2−2Vertex (3,−2), focus (3,−6), directrix y=2

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

How to do these

Find the vertex, focus and directrix of y=112(x+1)2+3

  1. The vertex is (-1, 3).
  2. 1/12 = 1/(4p), so 4p = 12 and p = 3.
  3. It opens up, so the focus is 3 above the vertex and the directrix 3 below.

The answer is Vertex (−1,3), focus (−1,6), directrix y=0.

Where students go wrong

Taking the coefficient as p. In y = (1/12)(x + 1)² + 3 the coefficient is 1 over 4p, so p = 3 and the focus sits 3 units from the vertex, not 1/12 of a unit.

Also called finding the vertex or axis of symmetry.

What you can put on this worksheet

Find the vertex
Find the vertex of y=2x2−12x+10 → (3,−8)
From the focus and directrix
Write the equation of the parabola with focus (0,2) and directrix y=0 → y=14x2+1
Focus and directrix from the equation
Find the vertex, focus and directrix of x=112y2 → Vertex (0,0), focus (3,0), directrix x=−3
Sideways, from general form
Find the vertex, focus and directrix: y2−2y+8x+1=0 → Vertex (0,1)
Focus (−2,1)
Directrix x=2
Vertex and a fractional directrix
Write the equation of the parabola with vertex (0,1) and directrix y=74 → y=−13x2+1
Vertex and an x-intercept
Write the equation, in vertex form, of the parabola with an x-intercept at 3 and vertex (0,18) → y=−2x2+18

Questions about these worksheets

Yes. Every vertex of a parabola 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 vertex of a parabola 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 6: find the vertex, from the focus and directrix, focus and directrix from the equation, sideways, from general form, vertex and a fractional directrix and vertex and an x-intercept. Tick as many as you want and set how many of each, or let it spread them evenly.

Reading the vertex straight off the standard form. In y = 2x² - 12x + 13 the vertex is not (12, 13) or (-12, 13); the equation has to be rearranged before anything can be read from it.