In a quadratic function y = ax^2 + bx + c, what must be true about the coefficient a to ensure a parabola?

In a quadratic function y = ax^2 + bx + c, what must be true about the coefficient a to ensure a parabola?

• A. a must be nonzero
• B. a must be zero
• C. a can be any real number
• D. a must be 1

Answer: (A)

A parabola comes from a quadratic function because the x^2 term creates the second-degree shape. That x^2 term must have a nonzero coefficient; if a were zero, the equation becomes y = bx + c, which is a straight line, not a parabola. So a must be nonzero. Any nonzero real value works, with the sign of a determining whether the parabola opens upward (a > 0) or downward (a < 0), and the magnitude of a affecting how narrow or wide the curve is.

A parabola comes from a quadratic function because the x^2 term creates the second-degree shape. That x^2 term must have a nonzero coefficient; if a were zero, the equation becomes y = bx + c, which is a straight line, not a parabola. So a must be nonzero. Any nonzero real value works, with the sign of a determining whether the parabola opens upward (a > 0) or downward (a < 0), and the magnitude of a affecting how narrow or wide the curve is.