Quick answer
A quadratic y = ax² + bx + c has vertex (h, k) with h = −b ÷ 2a and k = c − b² ÷ 4a, giving vertex form y = a(x − h)² + k. For y = x² − 4x + 7 the vertex is (2, 3), so y = 1(x − 2)² + 3.
Enter a, b and c. The calculator returns the vertex and the vertex form.
Related tools: Slope-Intercept Form Calculator: Line equation y = mx + b from a slope and a point. Point-Slope Form Calculator: Point-slope and slope-intercept equations of a line. Standard Form to Slope-Intercept Form Calculator: Convert Ax + By = C to y = mx + b.
How to use the Vertex Form Calculator
- Enter a.
- Enter b.
- Enter c.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| a | A number | — |
| b | A number | — |
| c | A number | — |
Formula and method
Formula
h = −b ÷ 2a; k = c − b² ÷ 4a; y = a(x − h)² + k
IEEE-754 double precision; shown to at most 6 decimal places.
Completing the square.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: a = 1; b = -4; c = 7.
| Result | Value |
|---|---|
| Vertex (h, k) | (2, 3) |
| Form | y = 1(x − 2)² + 3 |
Example 2
Inputs: a = 2; b = 4; c = 1.
| Result | Value |
|---|---|
| Vertex (h, k) | (-1, -1) |
| Form | y = 2(x + 1)² − 1 |
The first line is the headline result (vertex (h, k)). Below it the vertex Form Calculator lists form, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Completing the square
x² − 4x + 7 = (x − 2)² − 4 + 7 = (x − 2)² + 3. The parabola opens upward (a > 0), so the vertex (2, 3) is the minimum point.
Reading the form
h shifts the parabola horizontally and k vertically; a stretches it and flips it when negative. The coefficient a must be non-zero.
How the result changes with a
Holding every other input at the first worked example, this table changes only a. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| a | Vertex (h, k) | Form |
|---|---|---|
| 0.5 | (4, -1) | y = 0.5(x − 4)² − 1 |
| 1 | (2, 3) | y = 1(x − 2)² + 3 |
| 2 | (1, 5) | y = 2(x − 1)² + 5 |
| 4 | (0.5, 6) | y = 4(x − 0.5)² + 6 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- a: entering "0" shows "a must not be 0." beside that field and no result.
Choosing between the Vertex Form Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Slope-Intercept Form Calculator](/math/slope-intercept-form/) | slope, point x and point y | Equation |
| [Point-Slope Form Calculator](/math/point-slope-form/) | slope, point x₁ and point y₁ | Point-slope form |
| [Standard Form to Slope-Intercept Form Calculator](/math/standard-form-to-slope-intercept-form/) | a, b and c | Slope-intercept form |
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
| [LCM Calculator](/math/lcm-calculator/) | numbers | Least common multiple |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- a ≠ 0
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for a to confirm this.
Frequently asked questions
What is the vertex of y = −x² + 4x?
For y = −x² + 4x, h = −4 ÷ −2 = 2 and k = 4, so the vertex (2, 4) is a maximum.
Is the vertex a maximum or minimum?
A minimum if a > 0, a maximum if a < 0.
What is the axis of symmetry?
The axis of symmetry is the vertical line x = h through the vertex.
What does the Vertex Form Calculator calculate?
Convert y = ax² + bx + c to vertex form. It returns Vertex (h, k) and Form.
What formula does the Vertex Form Calculator use?
h = −b ÷ 2a; k = c − b² ÷ 4a; y = a(x − h)² + k. Completing the square.
What do I need to enter in the vertex Form Calculator?
You enter a, b and c.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. a ≠ 0.
What is an example of a vertex Form result?
With a 1, b -4 and c 7, the vertex (h, k) is (2, 3).
Related calculators
- Slope-Intercept Form Calculator — Line equation y = mx + b from a slope and a point.
- Point-Slope Form Calculator — Point-slope and slope-intercept equations of a line.
- Standard Form to Slope-Intercept Form Calculator — Convert Ax + By = C to y = mx + b.
- Slope Calculator — Slope, angle, distance and line equation through two points.
- LCM Calculator — Least common multiple of two or more numbers.
- Factor Calculator — List every factor of a positive whole number.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.