1. What is a Parabola Function?

A parabola is a U-shaped curve that can open upwards or downwards. It's the graph of a quadratic function in the form y = a(x - h)² + k, where (h,k) is the vertex 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)
• \( h \) — X-coordinate of the vertex
• \( k \) — Y-coordinate of the vertex
• \( x \) — Input value to evaluate the function at

Explanation: The equation shows how any point (x,y) on the parabola relates to its vertex and shape characteristics.

3. Importance of Parabola Calculations

Details: Parabolas are fundamental in physics (projectile motion), engineering (structural design), economics (profit curves), and many other fields.

4. Using the Calculator

Tips: Enter the coefficient 'a', vertex coordinates (h,k), and the x-value where you want to evaluate the function. The calculator will compute the corresponding y-value and parabola characteristics.

5. Frequently Asked Questions (FAQ)

Q1: What does the 'a' coefficient represent?
A: 'a' determines the parabola's width and direction. Larger |a| means narrower parabola. Positive 'a' opens upwards, negative opens downwards.

Q2: How do I find the roots of the parabola?
A: Set y=0 and solve for x: 0 = a(x - h)² + k. This gives x = h ± √(-k/a) when -k/a ≥ 0.

Q3: What's special about the vertex?
A: The vertex is the parabola's maximum (if a < 0) or minimum (if a > 0) point, and the curve is symmetric about the vertical line through it.

Q4: Can this represent all quadratic functions?
A: Yes, any quadratic y = ax² + bx + c can be rewritten in vertex form by completing the square.

Q5: How is this different from linear functions?
A: Parabolas have curvature (second-degree term) while linear functions are straight lines (first-degree only).