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

In this article, we will learn about finding the vertex, focus and directrix of a parabola using Python. A parabola is a U-shaped curve that can be represented by the standard form equation y = ax² + bx + c.

Understanding Parabola Components

The vertex is the point where the parabola changes direction (minimum or maximum point).

The focus is a point inside the parabola from which distances to any point on the curve have special properties.

The directrix is a horizontal line outside the parabola used to define the curve geometrically.

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):
    # 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 = vertex_y + 1 / (4 * a)
    
    # Calculate directrix equation
    directrix_y = vertex_y - 1 / (4 * a)
    
    print(f"Parabola: y = {a}x² + {b}x + {c}")
    print(f"Vertex: ({vertex_x:.4f}, {vertex_y:.4f})")
    print(f"Focus: ({focus_x:.4f}, {focus_y:.4f})")
    print(f"Directrix: y = {directrix_y:.4f}")

# Example usage
a = 1
b = -4
c = 3

find_parabola_properties(a, b, c)

Parabola: y = 1x² + -4x + 3
Vertex: (2.0000, -1.0000)
Focus: (2.0000, -0.7500)
Directrix: y = -1.2500

Another Example

Let's try with different coefficients ?

def find_parabola_properties(a, b, c):
    vertex_x = -b / (2 * a)
    vertex_y = (4 * a * c - b * b) / (4 * a)
    
    focus_x = vertex_x
    focus_y = vertex_y + 1 / (4 * a)
    
    directrix_y = vertex_y - 1 / (4 * a)
    
    print(f"Parabola: y = {a}x² + {b}x + {c}")
    print(f"Vertex: ({vertex_x:.4f}, {vertex_y:.4f})")
    print(f"Focus: ({focus_x:.4f}, {focus_y:.4f})")
    print(f"Directrix: y = {directrix_y:.4f}")

# Example with different values
a = 2
b = -8
c = 6

find_parabola_properties(a, b, c)

Parabola: y = 2x² + -8x + 6
Vertex: (2.0000, -2.0000)
Focus: (2.0000, -1.8750)
Directrix: y = -2.1250

Complete Program with Input Validation

def find_parabola_properties(a, b, c):
    if a == 0:
        print("Error: 'a' cannot be zero. This would not be a parabola.")
        return
    
    # Calculate properties
    vertex_x = -b / (2 * a)
    vertex_y = (4 * a * c - b * b) / (4 * a)
    
    focus_x = vertex_x
    focus_y = vertex_y + 1 / (4 * a)
    
    directrix_y = vertex_y - 1 / (4 * a)
    
    # Display results
    print(f"\nParabola equation: y = {a}x² + {b}x + {c}")
    print(f"Vertex: ({vertex_x:.4f}, {vertex_y:.4f})")
    print(f"Focus: ({focus_x:.4f}, {focus_y:.4f})")
    print(f"Directrix: y = {directrix_y:.4f}")
    
    # Determine if parabola opens upward or downward
    direction = "upward" if a > 0 else "downward"
    print(f"Parabola opens: {direction}")

# Test with multiple examples
examples = [(1, -2, 1), (2, 4, 1), (-1, 2, 3)]

for a, b, c in examples:
    find_parabola_properties(a, b, c)
    print("-" * 40)

Parabola equation: y = 1x² + -2x + 1
Vertex: (1.0000, 0.0000)
Focus: (1.0000, 0.2500)
Directrix: y = -0.2500
Parabola opens: upward
----------------------------------------

Parabola equation: y = 2x² + 4x + 1
Vertex: (-1.0000, -1.0000)
Focus: (-1.0000, -0.8750)
Directrix: y = -1.1250
Parabola opens: upward
----------------------------------------

Parabola equation: y = -1x² + 2x + 3
Vertex: (1.0000, 4.0000)
Focus: (1.0000, 3.7500)
Directrix: y = 4.2500
Parabola opens: downward
----------------------------------------

Conclusion

This program calculates the vertex, focus, and directrix of a parabola using the standard quadratic equation coefficients. The vertex represents the turning point, while the focus and directrix help define the parabola's geometric properties.