1. What is a Parabola?

A parabola is a U-shaped curve that is the graph of a quadratic function of the form y = ax² + bx + c. It has important properties like a vertex, axis of symmetry, and roots that are useful in physics, engineering, and mathematics.

2. How Does the Calculator Work?

The calculator uses the 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 Calculations:

• Vertex: \( x = -\frac{b}{2a} \), then find y
• Roots: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Y-intercept: \( (0, c) \)

3. Importance of Parabola Calculations

Details: Understanding parabola properties is essential for solving problems in projectile motion, optics (parabolic mirrors), economics, and many other fields.

4. Using the Calculator

Tips: Enter coefficients a, b, and c from your quadratic equation. The calculator will determine the vertex, y-intercept, and roots (if they exist).

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 (b² - 4ac) is negative.

Q2: What does a negative 'a' value mean?
A: A negative 'a' means the parabola opens downward instead of upward.

Q3: What is the vertex of a parabola?
A: The vertex is the highest or lowest point on the parabola (depending on whether it opens down or up).

Q4: Can I use this for linear equations?
A: No, this is specifically for quadratic equations (a ≠ 0). For linear equations (a = 0), use a different calculator.

Q5: How precise are the calculations?
A: Results are rounded to 2 decimal places for clarity, but calculations use full precision internally.