Vertex of a Parabola Calculator
Vertex of a Parabola Calculator
Enter the coefficients a, b, and c from your parabola's standard-form equation (y = ax² + bx + c). This calculates the vertex — the parabola's highest or lowest point — and converts the equation into vertex form.
Once you have the vertex coordinates (h, k), rewriting the equation as y = a(x − h)² + k makes the vertex's location immediately visible just by reading the equation, rather than needing to compute it separately every time. Whether the vertex represents a minimum or a maximum depends entirely on the sign of a: a positive a opens the parabola upward (a minimum at the vertex), while a negative a opens it downward (a maximum at the vertex) — a distinction that matters directly in optimization problems, like finding the maximum height of a projectile or the minimum cost on a cost curve.
- Formula: vertex x-coordinate h = −b ÷ 2a, and vertex y-coordinate k = c − b² ÷ 4a.
- Vertex form: y = a(x − h)² + k — once you have h and k, this form makes the vertex's location immediately visible in the equation itself.
- Minimum or maximum depends on the sign of a: when a is positive, the parabola opens upward and the vertex is the minimum point; when a is negative, it opens downward and the vertex is the maximum point.
How do I know if the vertex is a minimum or a maximum?
Check the sign of a — a positive a means the parabola opens upward like a "U" and the vertex is its lowest point, while a negative a means it opens downward and the vertex is its highest point.
Why is vertex form useful compared to standard form?
Vertex form directly shows the vertex coordinates (h, k) without any extra calculation, which is especially useful for graphing the parabola quickly or describing transformations like shifts.
Vertex of a Parabola Calculator
Enter the coefficients a, b, and c from your parabola's standard-form equation (y = ax² + bx + c). This calculates the vertex — the parabola's highest or lowest point — and converts the equation into vertex form.
Once you have the vertex coordinates (h, k), rewriting the equation as y = a(x − h)² + k makes the vertex's location immediately visible just by reading the equation, rather than needing to compute it separately every time. Whether the vertex represents a minimum or a maximum depends entirely on the sign of a: a positive a opens the parabola upward (a minimum at the vertex), while a negative a opens it downward (a maximum at the vertex) — a distinction that matters directly in optimization problems, like finding the maximum height of a projectile or the minimum cost on a cost curve.
- Formula: vertex x-coordinate h = −b ÷ 2a, and vertex y-coordinate k = c − b² ÷ 4a.
- Vertex form: y = a(x − h)² + k — once you have h and k, this form makes the vertex's location immediately visible in the equation itself.
- Minimum or maximum depends on the sign of a: when a is positive, the parabola opens upward and the vertex is the minimum point; when a is negative, it opens downward and the vertex is the maximum point.
How do I know if the vertex is a minimum or a maximum?
Check the sign of a — a positive a means the parabola opens upward like a "U" and the vertex is its lowest point, while a negative a means it opens downward and the vertex is its highest point.
Why is vertex form useful compared to standard form?
Vertex form directly shows the vertex coordinates (h, k) without any extra calculation, which is especially useful for graphing the parabola quickly or describing transformations like shifts.