1. What is a Parabola?

A parabola is a U-shaped curve that is the graph of a quadratic function. It's a conic section produced by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface.

2. How Does the Calculator Work?

The calculator plots the quadratic function:

Where:

• \( a \) — Determines the parabola's width and direction (up/down)
• \( b \) — Influences the position of the vertex and axis of symmetry
• \( c \) — The y-intercept of the parabola

Features Calculated: The calculator also determines the vertex (maximum/minimum point) and roots (x-intercepts) of the parabola.

3. Understanding the Results

Vertex: The highest or lowest point on the parabola. For \( y = ax^2 + bx + c \), the vertex is at \( x = -\frac{b}{2a} \).

Roots: The points where the parabola crosses the x-axis (if any). Calculated using the quadratic formula.

4. Using the Calculator

Tips: Enter coefficients a, b, and c. The calculator will plot the parabola and show key features. Try different values to see how they affect the shape and position.

5. Frequently Asked Questions (FAQ)

Q1: What if my parabola doesn't cross the x-axis?
A: This means there are no real roots (the discriminant is negative). The parabola is entirely above or below the x-axis.

Q2: What does a negative 'a' value do?
A: It makes the parabola open downward instead of upward.

Q3: What if a = 0?
A: The equation becomes linear (not quadratic), and you'll get a straight line instead of a parabola.

Q4: How can I make the parabola wider or narrower?
A: Smaller absolute values of 'a' make the parabola wider; larger values make it narrower.

Q5: What practical applications do parabolas have?
A: Parabolas are used in physics (projectile motion), engineering (satellite dishes, headlights), architecture, and many other fields.