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Parabola Vertex Calculator

Find the vertex of a parabola.

How it is calculated

For y = ax² + bx + c the vertex lies on the axis of symmetry, at x = -b / (2a). The y-coordinate comes from substituting that x back into the equation: y = a·x² + b·x + c. The coefficients a, b and c are plain numbers with no units. The parabola opens upward when a > 0, so the vertex is its minimum, and downward when a < 0, so the vertex is its maximum. If a is 0 the equation is a straight line with no vertex, and the tool returns no result. Both coordinates are rounded to two decimals for display.

Worked example

Take y = 2x² - 8x + 5, so a = 2, b = -8, c = 5. The x-coordinate is -(-8) / (2·2) = 8 / 4 = 2. Substituting: y = 2·2² - 8·2 + 5 = 8 - 16 + 5 = -3. The vertex is at x = 2.00, y = -3.00. Since a is positive, this is the lowest point of the curve.

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Common mistakes

  • Dropping the sign of b: the formula is -b/(2a), so b = -8 gives a positive x.
  • Dividing by a instead of 2a, which gives half the correct x.
  • Entering the equation before moving every term to one side, for example y = 3x² + 2x written as y - 3x² = 2x with the wrong coefficients. Write it as ax² + bx + c first; use c = 0 when the constant is missing.
  • Treating the vertex y-value as the x-intercept. The vertex is the extreme point, not a root.

About this calculator

This calculator finds the vertex of a parabola given in the form y = ax² + bx + c, using the vertex formula x = -b/(2a), then substituting back to get the y-coordinate. The vertex is the parabola's turning point — its minimum if a > 0, or its maximum if a < 0.

Algebra and precalculus students use it to check homework on quadratic functions, and it comes up in real applications like finding the maximum height of a thrown object, the optimal price point in a revenue model, or the lowest point of a suspension cable's curve.

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Comments (1)

  • Editorial team

    The sign of a tells you what the vertex means. With a negative a the parabola opens downward and the vertex is the highest point, which is what you want for the peak height of a thrown ball. With a positive a it is the lowest point, as in a minimum-cost curve. Try y = -5x² + 20x + 1: the vertex is at x = 2, y = 21. Tell us what problem you are modelling.

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