Vertex Calculator for Parabola $(h, k)$

Q: What is the vertex of a parabola?

The vertex is the highest or lowest point on the graph of a quadratic function (parabola). If the parabola opens upward ($a>0$), the vertex is the minimum point. If it opens downward ($a<0$), the vertex is the maximum point.

Q: What is the relationship between the vertex and the Axis of Symmetry?

The vertex always lies directly on the axis of symmetry. If the vertex is the point $(h, k)$, the axis of symmetry is the vertical line $x = h$.

Q: How does the 'a' value affect the vertex?

The sign of 'a' determines whether the vertex is a maximum or minimum. A positive 'a' (parabola opens up) means the vertex is a minimum. A negative 'a' (parabola opens down) means the vertex is a maximum. The magnitude of 'a' controls the width of the parabola but does not change the vertex coordinates themselves.

Q: What is the vertex in the context of optics or ophthalmology?

In optics, the vertex is often used to describe the **back vertex distance** or **front vertex distance** of a lens. This is the distance from the back or front surface of a spectacle lens to the cornea of the eye, which is critical for accurate contact lens fitting or high-power spectacle prescription grinding. This calculator focuses only on the mathematical vertex of a parabola.

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.

$y = ax^2 + bx + c$

a =

b =

c =

$y = a(x - h)^2 + k$

a =

h =

k =

$y = a(x - p)(x - q)$

a =

p =

q =

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:

1. X-coordinate ($h$):

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

2. Y-coordinate ($k$):

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

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:

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

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:

1. X-coordinate ($h$):

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

2. Y-coordinate ($k$):

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

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, $(-\infty, \infty)$.

Vertex Calculator for Parabola $(h, k)$

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Step-by-Step Calculation

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

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

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

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

Understanding the Vertex and Parabola Shape

Frequently Asked Questions (FAQ)

What is the vertex of a parabola?

What is the relationship between the vertex and the Axis of Symmetry?

How does the 'a' value affect the vertex?

What is the vertex in the context of optics or ophthalmology?