Parabola Properties Calculator

Analyze all the key properties of a parabola with step-by-step explanations

Equation: \(f(x) = x^2\)

Key Properties of a Parabola

A parabola has several important properties that help us understand its behavior:

  • Vertex: The highest or lowest point on the parabola
  • Axis of Symmetry: The vertical line passing through the vertex
  • Direction: Whether the parabola opens upward or downward
  • x-intercepts: Where the parabola crosses the x-axis
  • y-intercept: Where the parabola crosses the y-axis
  • Domain and Range: The set of possible x and y values
  • Minimum/Maximum Value: The y-coordinate of the vertex

Forms of a Parabola Equation

A parabola can be represented in different forms:

  • Standard Form: \(f(x) = ax^2 + bx + c\)
  • Vertex Form: \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex
  • Factored Form: \(f(x) = a(x - r_1)(x - r_2)\), where \(r_1\) and \(r_2\) are the roots

Each form makes certain properties of the parabola more apparent.

Applications of Parabolas

Parabolas have many real-world applications:

  • Physics: Projectile motion, reflection properties
  • Engineering: Bridges, arches, satellite dishes
  • Economics: Cost optimization, revenue models
  • Architecture: Parabolic structures for strength
  • Optics: Reflectors, telescopes, microscopes

Examples

Basic Parabola

Analyze the properties of \(f(x) = x^2\)

This is the simplest parabola:

  • Vertex: (0, 0)
  • Opens upward
  • Axis of symmetry: x = 0
  • y-intercept: (0, 0)
  • x-intercepts: (0, 0) only
  • Domain: All real numbers
  • Range: y ≥ 0

Parabola with Two x-intercepts

Analyze the properties of \(f(x) = x^2 - 4\)

This parabola has interesting properties:

  • Vertex: (0, -4)
  • Opens upward
  • Axis of symmetry: x = 0
  • y-intercept: (0, -4)
  • x-intercepts: (-2, 0) and (2, 0)
  • Domain: All real numbers
  • Range: y ≥ -4

Downward-Opening Parabola

Analyze the properties of \(f(x) = -2x^2 + 8x - 6\)

This parabola opens downward:

  • Vertex: (2, 2)
  • Opens downward
  • Axis of symmetry: x = 2
  • y-intercept: (0, -6)
  • x-intercepts: (1, 0) and (3, 0)
  • Domain: All real numbers
  • Range: y ≤ 2
  • Maximum value: 2 at x = 2

Frequently Asked Questions

How do I find the focus and directrix of a parabola?

For a parabola in the form f(x) = ax² + bx + c:

  1. First, convert to vertex form: f(x) = a(x - h)² + k
  2. The focus is located at (h, k + 1/(4a)) for a parabola that opens upward, or (h, k - 1/(4a)) for a parabola that opens downward
  3. The directrix is the horizontal line y = k - 1/(4a) for a parabola that opens upward, or y = k + 1/(4a) for a parabola that opens downward

The focus and directrix are related to the reflective properties of parabolas, which is why they're used in designing satellite dishes and telescopes.

What is the difference between a parabola's vertex and focus?

The vertex is the highest or lowest point on the parabola and is located at (h, k) in the vertex form f(x) = a(x - h)² + k.

The focus is a special point related to the reflective properties of the parabola. It's located at a distance of 1/(4|a|) from the vertex along the axis of symmetry. Any ray parallel to the axis of symmetry will reflect off the parabola and pass through the focus.

How do I find the equation of a parabola given its vertex and another point?

If you know the vertex (h, k) and another point (p, q) on the parabola:

  1. Start with the vertex form: f(x) = a(x - h)² + k
  2. Substitute the point (p, q): q = a(p - h)² + k
  3. Solve for a: a = (q - k)/((p - h)²)
  4. Substitute this value of a back into the vertex form to get the complete equation

If the vertex is (3, 2) and another point is (5, 10), then:

a = (10 - 2)/((5 - 3)²) = 8/4 = 2

So the equation is f(x) = 2(x - 3)² + 2

What is the relationship between a parabola's equation and its graph?

The equation of a parabola directly determines its graphical properties:

  • The coefficient 'a' determines whether the parabola opens upward (a > 0) or downward (a < 0), and how wide or narrow it is (larger |a| means narrower parabola)
  • The vertex (h, k) determines the position of the parabola's highest or lowest point
  • The x-intercepts, if they exist, are the solutions to ax² + bx + c = 0
  • The y-intercept is the value of f(0), which equals c in the standard form
  • The axis of symmetry is the vertical line x = h (or x = -b/(2a) in standard form)

Understanding these relationships helps in sketching parabolas accurately and interpreting their equations.