How to calculate the vertex of a parabola given three points

Calculating the Vertex of a Parabola from Three Given Points

Learn how to determine the vertex coordinates of a parabolic curve when provided with any three distinct points that lie on the parabola. This guide covers the mathematical principles and practical implementation.

Finding the vertex of a parabola is a common problem in mathematics, physics, and engineering. While the vertex form of a parabola (y = a(x-h)² + k) directly gives the vertex (h, k), you often encounter situations where the parabola is defined by three arbitrary points. This article will guide you through the process of deriving the parabolic equation from these points and subsequently calculating its vertex.

Understanding the Parabola Equation

A general parabolic equation can be expressed in the standard form: y = ax² + bx + c. Our goal is to find the coefficients a, b, and c using the three given points. Once these coefficients are known, we can use a simple formula to find the vertex.

Standard form of a parabola and its vertex

Step 1: Formulating a System of Equations

Given three distinct points (x1, y1), (x2, y2), and (x3, y3) that lie on the parabola, we can substitute each point's coordinates into the standard parabolic equation y = ax² + bx + c. This will yield a system of three linear equations with three unknowns (a, b, c):

1. y1 = ax1² + bx1 + c
2. y2 = ax2² + bx2 + c
3. y3 = ax3² + bx3 + c

System of linear equations from three points

Step 2: Solving for Coefficients a, b, and c

Solving this system of linear equations is the most computationally intensive part. You can use various methods such as substitution, elimination, or matrix methods (Cramer's Rule, Gaussian elimination). For programming, matrix methods are generally preferred for their robustness and ease of implementation.

import numpy as np

def solve_coefficients(p1, p2, p3):
    x1, y1 = p1
    x2, y2 = p2
    x3, y3 = p3

    # Construct the matrix A and vector B for Ax = B
    # where x = [a, b, c]
    A = np.array([
        [x1**2, x1, 1],
        [x2**2, x2, 1],
        [x3**2, x3, 1]
    ])
    B = np.array([y1, y2, y3])

    # Solve for a, b, c
    try:
        coefficients = np.linalg.solve(A, B)
        return coefficients
    except np.linalg.LinAlgError:
        print("Error: The points do not form a unique parabola (e.g., collinear points).")
        return None

# Example usage:
# p1 = (1, 2)
# p2 = (2, 3)
# p3 = (3, 6)
# a, b, c = solve_coefficients(p1, p2, p3)
# print(f"Coefficients: a={a}, b={b}, c={c}")

Python code to solve for coefficients a, b, c using NumPy

Step 3: Calculating the Vertex Coordinates

Once you have the values for a, b, and c, the vertex (h, k) of the parabola y = ax² + bx + c can be found using the following formulas:

h = -b / (2a)
k = a(h)² + b(h) + c  (or k = c - b² / (4a))

Formulas for the vertex coordinates (h, k)

Putting It All Together: Complete Example

Let's combine the steps into a single function to calculate the vertex given three points.

import numpy as np

def calculate_parabola_vertex(p1, p2, p3):
    """
    Calculates the vertex of a parabola given three points.
    Points should be tuples or lists of (x, y).
    """
    x1, y1 = p1
    x2, y2 = p2
    x3, y3 = p3

    # Step 1 & 2: Solve for coefficients a, b, c
    A = np.array([
        [x1**2, x1, 1],
        [x2**2, x2, 1],
        [x3**2, x3, 1]
    ])
    B = np.array([y1, y2, y3])

    try:
        a, b, c = np.linalg.solve(A, B)
    except np.linalg.LinAlgError:
        return None, "Error: Points are collinear or do not form a unique parabola."

    # Check if 'a' is effectively zero (not a parabola)
    if abs(a) < 1e-9: # Use a small epsilon for floating point comparison
        return None, "Error: 'a' coefficient is zero, indicating a straight line, not a parabola."

    # Step 3: Calculate vertex coordinates
    h = -b / (2 * a)
    k = a * (h**2) + b * h + c

    return (h, k), None

# Test cases
points1 = [(1, 2), (2, 3), (3, 6)] # Example from problem
vertex1, error1 = calculate_parabola_vertex(*points1)
if vertex1:
    print(f"Vertex for points {points1}: {vertex1}")
else:
    print(error1)

points2 = [(0, 0), (1, 1), (2, 4)] # y = x^2, vertex (0,0)
vertex2, error2 = calculate_parabola_vertex(*points2)
if vertex2:
    print(f"Vertex for points {points2}: {vertex2}")
else:
    print(error2)

points3 = [(0, 0), (1, 0), (2, 0)] # Collinear points
vertex3, error3 = calculate_parabola_vertex(*points3)
if vertex3:
    print(f"Vertex for points {points3}: {vertex3}")
else:
    print(error3)

Complete Python function to calculate parabola vertex