How to Calculate the Vertex of a Parabola

Parabola Vertex Calculator

Vertex Coordinates:

X-coordinate (h):

Y-coordinate (k):

(" + h.toFixed(4) + ", " + k.toFixed(4) + ")

A parabola is a U-shaped curve that is common in mathematics and physics, appearing in everything from the trajectory of a thrown ball to the design of satellite dishes. Understanding its properties, especially its vertex, is crucial for solving various problems.

What is a Parabola?

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It is defined as the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

The most common way to represent a parabola algebraically is through its standard quadratic equation form:

y = ax² + bx + c

where 'a', 'b', and 'c' are constant coefficients, and 'a' cannot be zero. The sign of 'a' determines the direction the parabola opens:

• If a > 0, the parabola opens upwards (like a U).
• If a < 0, the parabola opens downwards (like an inverted U).

Understanding the Vertex

The vertex of a parabola is its most extreme point. It's the point where the parabola changes direction. This means it's either the lowest point on the graph (if the parabola opens upwards) or the highest point on the graph (if the parabola opens downwards).

The vertex is a critical point because it represents the minimum or maximum value of the quadratic function. For example, in projectile motion, the vertex represents the maximum height reached by an object.

Formulas for Calculating the Vertex

For a parabola in the standard form y = ax² + bx + c, the coordinates of the vertex (h, k) can be found using two simple formulas:

1. X-coordinate of the Vertex (h):

   h = -b / (2a)

   This formula directly gives you the x-value of the vertex using the coefficients 'a' and 'b' from the quadratic equation.

2. Y-coordinate of the Vertex (k):

   Once you have the value of 'h', you can find the y-coordinate by substituting 'h' back into the original quadratic equation:

   k = a(h)² + b(h) + c

   This means k = f(h), where f(x) is the quadratic function.

Step-by-Step Calculation Example

Let's walk through an example to see how these formulas are applied.

Example 1: Find the vertex of the parabola y = x² - 4x + 3

1. Identify the coefficients:

   From the equation y = x² - 4x + 3, we have:

   • a = 1
   • b = -4
   • c = 3
2. Calculate the x-coordinate (h):

   Using the formula h = -b / (2a):

   h = -(-4) / (2 * 1)

   h = 4 / 2

   h = 2

3. Calculate the y-coordinate (k):

   Substitute h = 2 back into the original equation y = x² - 4x + 3:

   k = (2)² - 4(2) + 3

   k = 4 - 8 + 3

   k = -1

4. State the vertex:

   The vertex of the parabola y = x² - 4x + 3 is (2, -1).

   Since a = 1 (which is positive), this parabola opens upwards, and (2, -1) is its minimum point.

Example 2: Find the vertex of the parabola y = -2x² - 8x - 5

1. Identify the coefficients:
   • a = -2
   • b = -8
   • c = -5
2. Calculate the x-coordinate (h):

   h = -(-8) / (2 * -2)

   h = 8 / -4

   h = -2

3. Calculate the y-coordinate (k):

   k = -2(-2)² - 8(-2) - 5

   k = -2(4) + 16 - 5

   k = -8 + 16 - 5

   k = 3

4. State the vertex:

   The vertex of the parabola y = -2x² - 8x - 5 is (-2, 3).

   Since a = -2 (which is negative), this parabola opens downwards, and (-2, 3) is its maximum point.

Using the Parabola Vertex Calculator

Our online calculator simplifies this process. Just input the coefficients 'a', 'b', and 'c' from your quadratic equation y = ax² + bx + c into the respective fields. Click "Calculate Vertex," and the calculator will instantly provide the x and y coordinates of the parabola's vertex.

This tool is perfect for students, educators, and anyone needing to quickly find the vertex of a quadratic function without manual calculation.