The axis of symmetry of the parabola y = ax^2 + bx + c is x = -b/(2a). Which statement is true?

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Multiple Choice

The axis of symmetry of the parabola y = ax^2 + bx + c is x = -b/(2a). Which statement is true?

Explanation:
The axis of symmetry is the vertical line that passes through the vertex of the parabola. For a quadratic y = ax^2 + bx + c, the x-coordinate of the vertex is -b/(2a). This comes from completing the square: y = a[(x + b/(2a))^2] + (c - b^2/(4a)). The squared term reaches its minimum when x + b/(2a) = 0, so the vertex lies at x = -b/(2a). Equivalently, taking the derivative dy/dx = 2ax + b and setting it to zero gives the same x-coordinate, x = -b/(2a). Since this value depends on both a and b, the axis cannot be x = -a/(2b) or x = -c/(2a), and it is not independent of a. Therefore the axis is x = -b/(2a).

The axis of symmetry is the vertical line that passes through the vertex of the parabola. For a quadratic y = ax^2 + bx + c, the x-coordinate of the vertex is -b/(2a). This comes from completing the square: y = a[(x + b/(2a))^2] + (c - b^2/(4a)). The squared term reaches its minimum when x + b/(2a) = 0, so the vertex lies at x = -b/(2a). Equivalently, taking the derivative dy/dx = 2ax + b and setting it to zero gives the same x-coordinate, x = -b/(2a). Since this value depends on both a and b, the axis cannot be x = -a/(2b) or x = -c/(2a), and it is not independent of a. Therefore the axis is x = -b/(2a).

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