Parabola Calculator

Parabola Properties Calculator

Equation:

Vertex:

Axis of Symmetry:

Focus:

Directrix:

Opens:

Y-intercept:

Understanding the Parabola and Its Properties

A parabola is a U-shaped curve that is symmetrical and represents the graph of a quadratic function. It's a fundamental concept in mathematics and physics, appearing in everything from the trajectory of a projectile to the design of satellite dishes and car headlights.

The Standard Form of a Parabola

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

y = ax² + bx + c

Here, 'a', 'b', and 'c' are coefficients that determine the shape, position, and orientation of the parabola. The 'a' coefficient is particularly important: if a > 0, the parabola opens upwards; if a < 0, it opens downwards. If a = 0, the equation simplifies to y = bx + c, which is a straight line, not a parabola.

Key Properties of a Parabola

Every parabola has several defining characteristics:

1. Vertex (h, k): This is the turning point of the parabola. If the parabola opens upwards, the vertex is the lowest point (minimum); if it opens downwards, it's the highest point (maximum). The coordinates of the vertex can be calculated using the coefficients 'a' and 'b':
   • h = -b / (2a)
   • k = a(h)² + b(h) + c
2. Axis of Symmetry: This is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. Its equation is simply x = h.
3. Focus: The focus is a fixed point on the interior of the parabola. It's a crucial element in the definition of a parabola: every point on the parabola is equidistant from the focus and the directrix. For a parabola in the form y = ax² + bx + c, the focus is located at (h, k + 1/(4a)).
4. Directrix: The directrix is a fixed line outside the parabola. As mentioned, every point on the parabola is equidistant from the focus and the directrix. Its equation is y = k - 1/(4a).
5. Y-intercept: This is the point where the parabola crosses the y-axis. It occurs when x = 0, so the y-intercept is always (0, c).
6. X-intercept(s) (Roots): These are the points where the parabola crosses the x-axis (where y = 0). They are found by solving the quadratic equation ax² + bx + c = 0 using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).
   • If b² - 4ac > 0, there are two distinct x-intercepts.
   • If b² - 4ac = 0, there is exactly one x-intercept (the vertex touches the x-axis).
   • If b² - 4ac < 0, there are no real x-intercepts (the parabola does not cross the x-axis).

How to Use the Parabola Calculator

Our Parabola Properties Calculator simplifies the process of finding these key characteristics. Simply input the coefficients 'a', 'b', and 'c' from your parabola's equation (y = ax² + bx + c) into the respective fields. The calculator will instantly compute and display:

• The coordinates of the Vertex
• The equation of the Axis of Symmetry
• The coordinates of the Focus
• The equation of the Directrix
• The direction in which the parabola opens (upwards or downwards)
• The Y-intercept
• Any real X-intercepts

Example Calculation:

Let's consider the parabola with the equation: y = x² - 4x + 3

Here, a = 1, b = -4, and c = 3.

• Vertex (h, k):
  • h = -(-4) / (2 * 1) = 4 / 2 = 2
  • k = (1)(2)² + (-4)(2) + 3 = 4 - 8 + 3 = -1
  • Vertex: (2, -1)
• Axis of Symmetry: x = 2
• Focus:
  • Focus Y = -1 + (1 / (4 * 1)) = -1 + 0.25 = -0.75
  • Focus: (2, -0.75)
• Directrix:
  • Directrix Y = -1 - (1 / (4 * 1)) = -1 - 0.25 = -1.25
  • Directrix: y = -1.25
• Opens: Since a = 1 (which is > 0), it opens Upwards.
• Y-intercept: (0, 3)
• X-intercepts:
  • Discriminant = (-4)² - 4(1)(3) = 16 - 12 = 4
  • x = (4 ± √4) / (2 * 1) = (4 ± 2) / 2
  • x1 = (4 + 2) / 2 = 3
  • x2 = (4 - 2) / 2 = 1
  • X-intercepts: (1, 0) and (3, 0)

This calculator is an invaluable tool for students, educators, and anyone working with quadratic equations, providing quick and accurate insights into the geometry of parabolas.