1. What is a Parabola?

A parabola is a U-shaped curve that is the graph of a quadratic function. It has important applications in physics, engineering, and many other fields.

2. How Does the Calculator Work?

The calculator uses the standard quadratic equation:

Where:

• \( a \) — Determines the parabola's width and direction (up/down)
• \( b \) — Affects the position of the vertex
• \( c \) — The y-intercept point

Key Features Calculated:

• Vertex (the maximum or minimum point)
• Y-intercept (where the parabola crosses the y-axis)
• Roots/x-intercepts (where y=0, if they exist)

3. Importance of Parabola Calculations

Details: Understanding parabola properties is essential in physics (projectile motion), engineering (structural design), economics (profit maximization), and computer graphics.

4. Using the Calculator

Tips: Enter the coefficients a, b, and c from your quadratic equation. The calculator will determine key properties of the parabola.

5. Frequently Asked Questions (FAQ)

Q1: What if my parabola doesn't have real roots?
A: The calculator will indicate "No real roots" when the discriminant is negative (b² - 4ac < 0).

Q2: What does the vertex represent?
A: The vertex is the maximum or minimum point of the parabola, depending on whether it opens downward or upward.

Q3: How is the axis of symmetry determined?
A: The axis of symmetry is the vertical line x = -b/(2a), which passes through the vertex.

Q4: What does coefficient 'a' tell us?
A: If a > 0, the parabola opens upward; if a < 0, it opens downward. The absolute value of a affects how "wide" or "narrow" the parabola is.

Q5: Can I use this for vertex form or factored form?
A: This calculator uses the standard form. Convert other forms to standard form before entering coefficients.