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)
• \( b \) — Influences the position of the vertex
• \( c \) — The y-intercept of the parabola

Key Calculations:

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

3. Importance of Parabola Calculations

Details: Parabolas model many real-world phenomena like projectile motion, reflector shapes, and optimization problems. Knowing the vertex helps find maximum/minimum values, and roots show where the function crosses the x-axis.

4. Using the Calculator

Tips: Enter coefficients a, b, and c. Optionally enter an x value to calculate the corresponding y value. The calculator will show the vertex, discriminant, roots, and optional y value.

5. Frequently Asked Questions (FAQ)

Q1: What if the discriminant is negative?
A: A negative discriminant means there are no real roots (the parabola doesn't cross the x-axis).

Q2: What does a zero discriminant mean?
A: A discriminant of zero means there's exactly one real root (the vertex touches the x-axis).

Q3: How does coefficient 'a' affect the parabola?
A: If a > 0, the parabola opens upward. If a < 0, it opens downward. Larger absolute values of a make the parabola narrower.

Q4: What's special about the vertex?
A: The vertex is the maximum or minimum point of the parabola, depending on whether it opens downward or upward.

Q5: Can I use this for vertical parabolas?
A: This calculator is for standard y = ax² + bx + c form. For x = ay² + by + c forms, you'd need a different approach.