Vertex of a Parabola Calculator
Calculate the coordinates of the vertex (h, k) and axis of symmetry of a parabola from standard quadratic form y = axยฒ + bx + c.
Enter Calculation Inputs
Fill in the fields below. Instant client-side computation runs automatically in your browser as you type.
Common Searches & Natural Queries
What is the Parabola Vertex?
The vertex of a parabola represents its extreme valueโthe absolute minimum when the parabola opens upwards (a > 0) or the absolute maximum when it opens downwards (a < 0). In standard algebraic form y = axยฒ + bx + c, the vertex coordinates (h, k) reveal the turning point of the quadratic trajectory.
Mathematical Formula & Methodology
The mathematical model underlying this tool operates on the following verified equation:
h = -b / (2a), k = c - (bยฒ / 4a) = f(h)
Understanding the Parameters:
- Coefficient a (non-zero) โ Primary input variable.
- Coefficient b โ Primary input variable.
- Constant c โ Primary input variable.
How to Use This Calculator
Enter the quadratic coefficients a, b, and c into the corresponding input fields. Note that coefficient a cannot be zero (otherwise the equation describes a straight line, not a parabola). Click Calculate to instantly receive the vertex coordinates, the axis of symmetry equation, the direction of opening, and the discriminant.
Worked Numerical Example
Given the quadratic equation y = 2xยฒ - 8x + 6: Here a = 2, b = -8, c = 6. First find h: h = -(-8) / (2 ร 2) = 8 / 4 = 2. Then evaluate k: k = 2(2)ยฒ - 8(2) + 6 = 8 - 16 + 6 = -2. The vertex is located at (2, -2), with an axis of symmetry at x = 2 and a minimum value of -2.
Key Considerations & Standards
Ensure a โ 0. The discriminant ฮ = bยฒ - 4ac determines whether the parabola intersects the x-axis at two points (ฮ > 0), one point (ฮ = 0), or has no real roots (ฮ < 0).
Frequently Asked Questions
What is the vertex formula in standard form?
For y = axยฒ + bx + c, the x-coordinate of the vertex is h = -b / (2a). The y-coordinate k is found by substituting h back into the function: k = f(h).
Does a positive "a" value create a maximum or minimum?
A positive "a" opens upward (U-shaped), making the vertex a minimum point. A negative "a" opens downward, making the vertex a maximum point.