Math

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Vertex Form Calculator.

Evaluate y = a(x − h)² + k and report its vertex.

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Set your values

Results update as you type.

Value: 7

Value

7.000000
Vertex x: 3

Vertex x

3.000000
Vertex y: -1

Vertex y

-1.000000

Results update automatically as you type.

Use Cases

Graphing Quadratic Functions

Quickly determine the vertex and direction of a parabola to sketch its graph accurately without completing the square.

Example: For y = 2(x - 1)^2 + 3, the vertex is (1, 3) and it opens upward.

Solving Real-World Optimization Problems

Use vertex form to find maximum or minimum values in physics, economics, or engineering, such as projectile height or profit maximization.

Example: Modeling the height of a ball: h(t) = -16(t - 2)^2 + 64 gives maximum height 64 at t=2.

Frequently Asked Questions

What is vertex form of a quadratic equation?
Vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. The value of 'a' determines the direction and width of the parabola.
How do I find the vertex from the vertex form?
The vertex is simply the point (h, k). In the equation y = a(x - h)^2 + k, the vertex is at (h, k). For example, if h=3 and k=-2, the vertex is (3, -2).
What does the value of 'a' tell me?
If 'a' is positive, the parabola opens upward; if negative, it opens downward. The absolute value of 'a' affects the width: larger |a| makes it narrower, smaller |a| makes it wider.

Tips & Common Mistakes

Tips

  • Ensure you input the correct signs: the vertex form is y = a(x - h)^2 + k, so if your equation has (x + 3), then h = -3.
  • Use the vertex coordinates to quickly identify the axis of symmetry: x = h.
  • Check the sign of 'a' to know if the vertex is a maximum (a < 0) or minimum (a > 0).
  • If you have a standard form equation, convert it to vertex form by completing the square before using this calculator.

Common Mistakes to Avoid

  • Forgetting to change the sign of h when entering the value. For (x - 2)^2, h is 2, but for (x + 2)^2, h is -2.
  • Confusing the vertex coordinates: the vertex is (h, k), not (a, h) or (a, k).
  • Assuming the vertex is always a minimum; it is a maximum if 'a' is negative.

Last updated: August 13, 2026