1. What is the Parabola Equation?

The parabola equation \( y = a x^2 + b x + c \) describes a quadratic function that graphs as a U-shaped curve (parabola) in the coordinate plane. The coefficients a, b, and c determine the shape and position of the parabola.

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
• \( c \) — y-intercept of the parabola
• \( x \) — Input value to calculate corresponding y

Additional Calculations:

• Vertex: \( \left( \frac{-b}{2a}, f\left(\frac{-b}{2a}\right) \right) \)
• Discriminant: \( b^2 - 4ac \) (determines number of roots)

3. Importance of Parabola Calculation

Details: Parabolas are fundamental in physics (projectile motion), engineering (structural design), economics (profit maximization), and computer graphics.

4. Using the Calculator

Tips: Enter coefficients a, b, c and an x value to calculate the corresponding y value. The calculator also provides the vertex and discriminant.

5. Frequently Asked Questions (FAQ)

Q1: What does the coefficient 'a' determine?
A: If a > 0, parabola opens upward; if a < 0, it opens downward. The absolute value of a affects the parabola's width.

Q2: How is the vertex important?
A: The vertex represents the maximum or minimum point of the parabola, crucial in optimization problems.

Q3: What does the discriminant tell us?
A: The discriminant indicates the number of x-intercepts: positive (2 roots), zero (1 root), negative (no real roots).

Q4: Can a be zero?
A: If a = 0, the equation becomes linear (y = bx + c), not quadratic. The calculator will still work but won't calculate vertex.

Q5: How are parabolas used in real life?
A: Parabolas model projectile trajectories, satellite dishes, suspension bridge cables, and optimal business pricing strategies.