How to Calculate the Vertex of a Parabola

Parabola Vertex Calculator

function calculateVertex() { var a = parseFloat(document.getElementById('coefficientA').value); var b = parseFloat(document.getElementById('coefficientB').value); var c = parseFloat(document.getElementById('coefficientC').value); var resultDiv = document.getElementById('vertexResult'); if (isNaN(a) || isNaN(b) || isNaN(c)) { resultDiv.innerHTML = "Please enter valid numbers for all coefficients."; resultDiv.style.color = "#dc3545"; return; } if (a === 0) { resultDiv.innerHTML = "Coefficient 'a' cannot be zero for a parabola. This would be a straight line."; resultDiv.style.color = "#dc3545"; return; } // Calculate x-coordinate of the vertex (h) var h = -b / (2 * a); // Calculate y-coordinate of the vertex (k) // k = a*h^2 + b*h + c var k = a * Math.pow(h, 2) + b * h + c; resultDiv.innerHTML = "The vertex of the parabola is: (" + h.toFixed(4) + ", " + k.toFixed(4) + ")"; resultDiv.style.color = "#28a745"; } // Initial calculation on page load for default values window.onload = calculateVertex;

A parabola is a U-shaped curve that is the graphical representation of a quadratic equation. These equations are fundamental in mathematics and appear in various real-world applications, from the trajectory of a projectile to the design of satellite dishes and bridge arches. The most crucial point on a parabola is its vertex.

What is a Parabola?

In its simplest form, a parabola is defined by a quadratic equation, which can be written in the standard form:

y = ax² + bx + c

Here, 'a', 'b', and 'c' are coefficients, where 'a' cannot be zero. If 'a' were zero, the equation would become y = bx + c, which is the equation of a straight line, not a parabola.

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

The Significance of the Vertex

The vertex is the turning point of the parabola. It is the point where the parabola changes direction. Depending on whether the parabola opens upwards or downwards:

• If the parabola opens upwards (a > 0), the vertex represents the minimum point of the function.
• If the parabola opens downwards (a < 0), the vertex represents the maximum point of the function.

This makes the vertex incredibly important for optimization problems, such as finding the maximum height reached by a ball or the minimum cost in a business model.

How to Calculate the Vertex

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

1. Calculating the x-coordinate (h)

The x-coordinate of the vertex, often denoted as 'h', is given by the formula:

h = -b / (2a)

This formula is derived from calculus (finding where the derivative is zero) or by using the midpoint of the parabola's roots.

2. Calculating the y-coordinate (k)

Once you have the x-coordinate (h), you can find the y-coordinate of the vertex, often denoted as 'k', by substituting 'h' back into the original quadratic equation:

k = a(h)² + b(h) + c

This means 'k' is simply the function's value at 'h', or f(h).

Step-by-Step Example

Let's find the vertex of the parabola given by the equation: y = 2x² - 8x + 6

Step 1: Identify the coefficients.
From the equation y = 2x² - 8x + 6, we have:

• a = 2
• b = -8
• c = 6

Step 2: Calculate the x-coordinate (h).
Using the formula h = -b / (2a):

h = -(-8) / (2 * 2)
h = 8 / 4
h = 2

Step 3: Calculate the y-coordinate (k).
Substitute h = 2 back into the original equation y = 2x² - 8x + 6:

k = 2(2)² - 8(2) + 6
k = 2(4) - 16 + 6
k = 8 - 16 + 6
k = -8 + 6
k = -2

Step 4: State the vertex.
The vertex of the parabola y = 2x² - 8x + 6 is (2, -2).

Since a = 2 (which is greater than 0), this parabola opens upwards, and the vertex (2, -2) is its minimum point.

Use the calculator above to quickly find the vertex for any quadratic equation by simply entering its 'a', 'b', and 'c' coefficients!