1. What is a Parabola?

A parabola is a U-shaped curve that is the graph of a quadratic function. It's defined by the equation y = ax² + bx + c, where a, b, and c are coefficients that determine its shape and position.

2. How Does the Calculator Work?

The calculator uses the standard parabola equation:

Where:

• \( a \) — Determines the parabola's width and direction (up/down)
• \( b \) — Influences the position of the vertex
• \( c \) — The y-intercept of the parabola
• \( x \) — The input value for which y is calculated

Additional Calculations:

• Vertex: \( x = -\frac{b}{2a} \), then calculate y
• Discriminant: \( D = b^2 - 4ac \) (determines roots)

3. Importance of Parabola Calculation

Details: Parabolas are fundamental in physics (projectile motion), engineering (structural design), economics (profit curves), and many other fields. Understanding their properties helps in modeling real-world phenomena.

4. Using the Calculator

Tips: Enter coefficients a, b, c and the x value. The calculator will compute the corresponding y value, vertex coordinates, and discriminant. For a ≠ 0.

5. Frequently Asked Questions (FAQ)

Q1: What does the discriminant tell us?
A: The discriminant (D) indicates the nature of the roots: D > 0 (two real roots), D = 0 (one real root), D < 0 (no real roots).

Q2: What happens when a is positive vs negative?
A: Positive a opens upward, negative a opens downward. The absolute value of a affects how "wide" or "narrow" the parabola appears.

Q3: What is the vertex of a parabola?
A: The vertex is the highest or lowest point on the parabola (depending on direction). It's where the curve changes direction.

Q4: Can a be zero in a parabola?
A: No, if a = 0, the equation becomes linear (y = bx + c). A parabola requires a quadratic term (x²).

Q5: How are parabolas used in real life?
A: Parabolas model projectile trajectories, satellite dishes use parabolic shapes to focus signals, and car headlights use parabolic reflectors.