What Is a Vertex Calculator?
A vertex calculator finds the vertex of a parabola — the highest or lowest point of a quadratic function. The vertex is the turning point of the curve and is key to graphing, optimization, and understanding the function's range. Enter the coefficients of your quadratic and the calculator returns the vertex coordinates, the axis of symmetry, and whether the vertex is a maximum or minimum.
How to Use the Calculator
- Enter coefficients a, b, and c from y = ax² + bx + c.
- Calculate — see the vertex, axis of symmetry, and max/min.
The Vertex Formula
The x-coordinate of the vertex is:
x = −b ÷ (2a)
Substitute this back into the equation to find the y-coordinate. The result (x, y) is the vertex. For example, y = x² − 4x + 3 gives x = 4 ÷ 2 = 2, and y = 4 − 8 + 3 = −1, so the vertex is (2, −1).
Maximum or Minimum?
| Coefficient a | Parabola Opens | Vertex Is |
|---|---|---|
| a > 0 | Upward | Minimum point |
| a < 0 | Downward | Maximum point |
Vertex Form
A quadratic can be written in vertex form y = a(x − h)² + k, where (h, k) is the vertex. This form makes the vertex obvious and shows how the parabola is shifted and stretched. Completing the square converts standard form into vertex form.
Why the Vertex Matters
- Optimization: the vertex gives the maximum or minimum value.
- Graphing: it anchors the shape of the parabola.
- Range: the vertex's y-value sets the function's range.
Frequently Asked Questions
How do you find the vertex of a parabola?
Find x = −b ÷ (2a), then substitute it back into the equation to get the y-coordinate. The pair (x, y) is the vertex.
Is the vertex a maximum or minimum?
If a is positive the parabola opens upward and the vertex is a minimum; if a is negative it opens downward and the vertex is a maximum.
What is vertex form?
Vertex form is y = a(x − h)² + k, where (h, k) is the vertex. It makes the turning point and transformations easy to read.
How is the vertex related to the axis of symmetry?
The axis of symmetry is the vertical line x = −b/(2a), which passes exactly through the vertex.
Is this vertex calculator free?
Yes — it is completely free, requires no signup, and finds the vertex and axis of symmetry.