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Parabola Equation:
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A parabola is a U-shaped curve that is the graph of a quadratic function. The standard form of a parabola equation is y = ax² + bx + c, where a, b, and c are coefficients that determine the shape and position of the parabola.
The calculator uses the quadratic equation:
Where:
Additional Calculations: The calculator also finds the vertex (h,k) using h = -b/(2a) and k = f(h), and determines roots using the quadratic formula.
Details: Parabolas are fundamental in physics (projectile motion), engineering (structural design), economics (profit maximization), and many other fields. Understanding their properties is essential for modeling quadratic relationships.
Tips: Enter the coefficients a, b, c and the x value for which you want to calculate y. The calculator will also show the vertex and roots of the parabola.
Q1: What does the 'a' coefficient represent?
A: The 'a' determines how "wide" or "narrow" the parabola is and whether it opens upward (a > 0) or downward (a < 0).
Q2: How is the vertex calculated?
A: The vertex x-coordinate is at x = -b/(2a), and the y-coordinate is found by plugging this x back into the equation.
Q3: What determines if a parabola has real roots?
A: The discriminant (b² - 4ac) determines the nature of the roots. If positive, two real roots; if zero, one real root; if negative, no real roots.
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: What are practical applications of parabolas?
A: Parabolas model projectile trajectories, optimize profit in business, design satellite dishes and headlights, and appear in architectural structures.