Vertex Form Solver
The vertex form of a quadratic, y = a(x − h)² + k, puts the most useful information on display: the vertex (h, k), the direction the parabola opens and how steep it is. This page shows how to convert any quadratic to vertex form, works through examples, and describes a pack of printable practice worksheets with complete answer keys.
- Instant download
- Editable Word & Excel
- Print-ready PDF
- 7-day refund
7 files · PDF, DOCX, XLSX
Vertex Form Worksheet Pack
Six vertex form worksheets (60 problems) with full answer keys, parabola graph paper, a formula reference card and an Excel converter for checking.
- Standard → vertex (PDF/DOCX)
- Vertex → standard (PDF/DOCX)
- Vertex, axis, range (PDF/DOCX)
- + 4 more
$5.00 USD, one-time 7 files · about $0.71 each
- Instant download
- 7-day refund
- Secure checkout by Stripe
7-day money-back guarantee. Full refund within 7 days of purchase, even after you download.
Free sample: Standard to Vertex Form (PDF, 100 KB)
Vertex Form Worksheet Pack
Six vertex form worksheets (60 problems) with full answer keys, parabola graph paper, a formula reference card and an Excel converter for checking.
- Standard → vertex (PDF/DOCX)
- Vertex → standard (PDF/DOCX)
- Vertex, axis, range (PDF/DOCX)
- Intercepts (PDF/DOCX)
- Reference card (PDF)
- Excel converter (XLSX)
- Graph paper (PDF)
Formats: PDF, DOCX, XLSX. Instant download after payment (link valid 72 hours, up to 5 downloads). AI-assisted: the templates were drafted with AI help and reviewed and laid out before publishing.
$5.00 USD, one-time 7 files · about $0.71 each
- Instant download
- 7-day refund
- Secure checkout by Stripe
7-day money-back guarantee. Full refund within 7 days of purchase, even after you download.
Free sample: Standard to Vertex Form (PDF, 100 KB)
Full refund within 7 days — see the refund policy and license.
Free printables
Single sheets you can print right now, no sign-up. The full pack adds every sheet in one download plus the editable files.
Vertex Form Completing the Square Worksheet2 pages · free PDF
Vertex Form to Standard Form Worksheet2 pages · free PDF
Printable Vertex Form Worksheet2 pages · free PDF
Printable Vertex Form Intercepts Worksheet2 pages · free PDF
Printable Vertex Form Reference Card1 page · free PDF
Printable Vertex Form Solver Worksheet2 pages · free PDF
Printable Parabola Graph Paper1 page · free PDFVertex form at a glance
| Part | Meaning |
|---|---|
| a | Opens up if a > 0, down if a < 0; |a| > 1 is narrower, |a| < 1 wider than y = x² |
| h | Horizontal shift — the vertex’s x-coordinate (note the minus sign: (x − 3) means h = 3) |
| k | Vertical shift — the vertex’s y-coordinate |
| Vertex | (h, k) — the minimum if a > 0, the maximum if a < 0 |
| Axis of symmetry | The vertical line x = h |
| Range | y ≥ k if a > 0; y ≤ k if a < 0 |
Converting from standard form by completing the square
- Start with y = ax² + bx + c.
- Factor a out of the first two terms: y = a(x² + (b/a)x) + c.
- Add and subtract (b/2a)² inside the brackets.
- Rewrite the perfect square: y = a(x + b/2a)² + c − b²/4a.
- So h = −b / 2a and k = c − b² / 4a.
Shortcut: h = −b/(2a), then find k by substituting h into the original equation.
Worked examples
| Standard form | Vertex form | Vertex | Opens |
|---|---|---|---|
| y = x² − 6x + 5 | y = (x − 3)² − 4 | (3, -4) | Up |
| y = 2x² + 8x + 3 | y = 2(x + 2)² − 5 | (-2, -5) | Up |
| y = −x² + 4x + 1 | y = −(x − 2)² + 5 | (2, 5) | Down |
| y = 3x² − 12x + 7 | y = 3(x − 2)² − 5 | (2, -5) | Up |
| y = 0.5x² + 2x − 4 | y = 0.5(x + 2)² − 6 | (-2, -6) | Up |
| y = -2x² − 8x − 5 | y = -2(x + 2)² + 3 | (-2, 3) | Down |
Take the first: y = x² − 6x + 5. Half of −6 is −3, and (−3)² = 9, so y = (x² − 6x + 9) − 9 + 5 = (x − 3)² − 4. The vertex is (3, −4), the axis is x = 3 and, because a = 1 > 0, the minimum value is −4.
Finding intercepts from vertex form
- y-intercept: set x = 0, giving y = ah² + k.
- x-intercepts: set y = 0, so (x − h)² = −k/a.
- If −k/a is negative, there are no real x-intercepts.
- Otherwise x = h ± √(−k/a).
For y = (x − 3)² − 4: (x − 3)² = 4, so x = 3 ± 2, giving x-intercepts at 1 and 5, and the y-intercept is 9 − 4 = 5 — matching the standard form’s c = 5.
From factored form
If a quadratic is given as y = a(x − p)(x − q), the axis of symmetry is halfway between the roots: h = (p + q) / 2. Substitute h to find k. For y = 2(x − 1)(x − 5), h = 3 and k = 2(2)(−2) = −8, so y = 2(x − 3)² − 8.
Real-world vertex problems
Vertex form answers “maximum” and “minimum” questions directly. A ball’s height h(t) = −4.9(t − 1.5)² + 13 reaches its maximum of 13 m at 1.5 seconds. A business profit model P(x) = −2(x − 40)² + 3,200 is maximised at 40 units, with a profit of 3,200. In each case the vertex is the answer.
What’s in the pack
- Six worksheets (60 problems): standard → vertex, vertex → standard, identifying vertex/axis/range, intercepts, factored → vertex and word problems
- Full answer keys with the completed-square step for every problem
- Parabola graph paper with axes
- A one-page reference card of formulas and transformations
- An Excel sheet that converts ax² + bx + c to vertex form for checking
Worksheets print on US Letter and A4; the spreadsheet opens in Excel and Google Sheets.
AI-assisted content
This page and worksheets were drafted with AI assistance and reviewed before publishing; answers were generated from the formulas and checked.
Frequently asked questions
What is vertex form?
y = a(x − h)² + k, where (h, k) is the vertex.
How do I find the vertex from standard form?
h = −b/(2a); substitute h into the equation to get k.
Why is it x − h?
The graph shifts right by h, so (x − 3) means h = +3.
How do I convert vertex form to standard form?
Expand the square and simplify.
How do I know if the vertex is a maximum or minimum?
Minimum if a > 0, maximum if a < 0.
Is this a subscription?
No, it is a one-time download.