Inputs
Enter the coefficients
For f(x) = ax² + bx + c, enter a, b, and c. The coefficient a must not be zero.
The graph and roots use the exact coefficients you enter. A negative discriminant is reported with complex roots.
Math / Quadratics
Vertex Form Calculator
Convert a quadratic to vertex form and find its vertex, axis, roots, and graph.
Vertex formParabolaAxis of symmetryRoots
Quadratic worksheet
Result
Vertex form
1(x - 2)^2 - 1
Vertex
(2, -1)
Axis of symmetry
x = 2
Opening direction
Opens upward
Y-intercept
3
Discriminant
4
Roots
3, 1
Parabola graph
Vertex: (2, -1)
Conversion steps
- 1
Write standard form
x^2 - 4x + 3Start with the quadratic f(x) = ax² + bx + c.
- 2
Find h
h = -b/(2a) = 2The x-coordinate of the vertex is h = −b/(2a).
- 3
Find k
k = f(h) = -1Substitute h into the original function to find the y-coordinate k.
- 4
Write vertex form
f(x) = 1(x - 2)^2 - 1Substitute h and k into f(x) = a(x − h)² + k.
- 5
Check the discriminant
Δ = b^2 - 4ac = 4The discriminant indicates whether the parabola has zero, one, or two real x-intercepts.
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About this calculator
Method, formulas, and limits.What this does
Converts a quadratic in standard form ax² + bx + c into vertex form a(x − h)² + k and reports the vertex, axis of symmetry, direction, y-intercept, discriminant, and roots.
Who it is for
Students studying quadratic functions, teachers checking completed-square work, and anyone comparing a parabola's algebraic form with its graph.
How it works
The vertex coordinates are calculated with h = −b/(2a) and k = f(h). The discriminant b² − 4ac determines whether the x-intercepts are two real roots, one repeated root, or a complex pair.
Limitations
The calculator uses a deterministic real-number model and reports complex roots symbolically when the discriminant is negative. It does not model measurement uncertainty or fit a curve to data.
Key calculations
- Vertex Coordinates
- For f(x) = ax² + bx + c, h = −b/(2a) and k = f(h). The vertex is (h, k).
- Vertex Form
- f(x) = a(x − h)² + k. The sign of a determines whether the parabola opens up or down.
- Discriminant
- Δ = b² − 4ac. Positive gives two real roots, zero gives one repeated root, and negative gives complex roots.
Reference ranges
- Opens Up
- When a is positive, the vertex is the minimum point and the parabola opens upward.
- Opens Down
- When a is negative, the vertex is the maximum point and the parabola opens downward.
- Real Intercepts
- A nonnegative discriminant means the graph touches or crosses the x-axis at real x-values.
How to use it
- 1.Enter a, b, and cProvide the three coefficients in f(x) = ax² + bx + c. The coefficient a must not be zero.
- 2.Read the vertex formThe calculator evaluates h and k, then writes the equivalent vertex form and axis of symmetry.
- 3.Check the rootsUse the discriminant and root output to see where the parabola crosses or misses the x-axis.
- 4.Inspect the graphThe plotted curve is centered on the calculated vertex and highlights the vertex point.
For ax² + bx + c, calculate h = −b/(2a), evaluate k = f(h), and substitute into a(x − h)² + k. This calculator displays each of those steps.
It is the vertical line through the vertex, x = h. Every point on the left side of the parabola has a matching point on the right side.
The discriminant predicts the number of real x-intercepts: two when it is positive, one repeated intercept when it is zero, and none when it is negative.
A negative discriminant means the parabola does not cross the real x-axis. The calculator reports the conjugate complex roots instead.
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