Vertex Calculator for Parabola $(h, k)$

The vertex is the minimum or maximum point of a parabola. Use this calculator to find the exact coordinates $(h, k)$ from your quadratic equation, regardless of whether it is in Standard, Vertex, or Intercept Form.

Formulas for Finding the Vertex $(h, k)$

The vertex of a parabola, $(h, k)$, can be found using different methods depending on the form of the quadratic equation:

1. Standard Form: $y = ax^2 + bx + c$

This is the most common method and involves two steps:

This $h$ value is also the equation for the **Axis of Symmetry** ($x = h$).

2. Vertex Form: $y = a(x - h)^2 + k$

In this form, the vertex coordinates are given directly:

3. Intercept Form: $y = a(x - p)(x - q)$

This form gives the x-intercepts, $p$ and $q$. The x-coordinate of the vertex ($h$) is the midpoint between these intercepts:

Understanding the Vertex and Parabola Shape

The sign of the leading coefficient, $a$, dictates the general shape and orientation of the parabola, and therefore the nature of the vertex:

• If $a > 0$ (positive), the parabola opens **upward** . The vertex is the **minimum** point of the function.
• If $a < 0$ (negative), the parabola opens **downward** . The vertex is the **maximum** point of the function.

The **Range** of the function is determined by the y-coordinate ($k$) of the vertex. For $a>0$, the range is $[k, \infty)$. For $a<0$, the range is $(-\infty, k]$. The **Domain** is always all real numbers, $(-\infty, \infty)$.

Frequently Asked Questions (FAQ)

','

h = \frac{-b}{2a}

k = f(h) = a(h)^2 + b(h) + c

\text{Vertex} = (h, k)

h = \frac{p + q}{2}

k = f(h) = a(h - p)(h - q)

1. X-coordinate ($h$): $h = \frac{-b}{2a}$ 2. Y-coordinate ($k$): $k = f(h) = a(h)^2 + b(h) + c$

$\text{Vertex} = (h, k)$

1. X-coordinate ($h$): $h = \frac{p + q}{2}$ 2. Y-coordinate ($k$): $k = f(h) = a(h - p)(h - q)$