Python Program for Finding the vertex, focus and directrix of a parabola

In this article, we will learn how to find the vertex, focus, and directrix of a parabola given its equation in standard form y = ax² + bx + c.

Problem Statement

Given a parabola equation in the standard form y = ax² + bx + c, we need to find:

• Vertex: The point where the parabola changes direction
• Focus: A fixed point inside the parabola
• Directrix: A fixed line outside the parabola

Mathematical Formulas

For a parabola y = ax² + bx + c:

• Vertex: (-b/(2a), (4ac - b²)/(4a))
• Focus: (-b/(2a), (4ac - b² + 1)/(4a))
• Directrix: y = (4ac - b² - 1)/(4a)

Python Implementation

def find_parabola_properties(a, b, c):
    """
    Find vertex, focus, and directrix of parabola y = ax^2 + bx + c
    """
    # Calculate vertex coordinates
    vertex_x = -b / (2 * a)
    vertex_y = (4 * a * c - b * b) / (4 * a)
    
    # Calculate focus coordinates  
    focus_x = vertex_x
    focus_y = (4 * a * c - b * b + 1) / (4 * a)
    
    # Calculate directrix equation
    directrix_y = (4 * a * c - b * b - 1) / (4 * a)
    
    print(f"Parabola: y = {a}x² + {b}x + {c}")
    print(f"Vertex: ({vertex_x:.3f}, {vertex_y:.3f})")
    print(f"Focus: ({focus_x:.3f}, {focus_y:.3f})")
    print(f"Directrix: y = {directrix_y:.3f}")

# Example usage
a = 1
b = -2  
c = 3

find_parabola_properties(a, b, c)

The output of the above code is ?

Parabola: y = 1x² + -2x + 3
Vertex: (1.000, 2.000)
Focus: (1.000, 2.250)
Directrix: y = 1.750

Multiple Examples

# Test with different parabolas
test_cases = [
    (1, 0, 0),    # y = x²
    (2, 4, 1),    # y = 2x² + 4x + 1
    (-1, 2, 3)    # y = -x² + 2x + 3 (opens downward)
]

for i, (a, b, c) in enumerate(test_cases, 1):
    print(f"\nExample {i}:")
    find_parabola_properties(a, b, c)

The output shows properties of different parabolas ?

Example 1:
Parabola: y = 1x² + 0x + 0
Vertex: (0.000, 0.000)
Focus: (0.000, 0.250)
Directrix: y = -0.250

Example 2:
Parabola: y = 2x² + 4x + 1
Vertex: (-1.000, -1.000)
Focus: (-1.000, -0.875)
Directrix: y = -1.125

Example 3:
Parabola: y = -1x² + 2x + 3
Vertex: (1.000, 4.000)
Focus: (1.000, 3.750)
Directrix: y = 4.250

Key Properties

Conclusion

The vertex, focus, and directrix are fundamental properties of any parabola. Using the standard form coefficients a, b, and c, we can calculate these properties with simple mathematical formulas. The vertex represents the turning point, while the focus and directrix help define the parabola's geometric shape.