Parabola Property: Analyzing Vertex Position on X-axis and Negative Values

Question

If the vertex of the parabola is on the x-axis, then the function is always negative except for the vertex point.

Step-by-Step Solution

To solve this problem, let's consider the characteristics of a quadratic function in vertex form:

In vertex form, a quadratic function is written as y=a(xh)2+k y = a(x-h)^2 + k . Given that the vertex is on the x-axis, the vertex point, (h,k)(h, k), has k=0 k = 0 . Therefore, the equation becomes y=a(xh)2 y = a(x-h)^2 .

We need to determine if the function is always negative except for the vertex point. This boils down to the sign of the coefficient a a :

  • If a<0 a < 0 , the parabola opens downward, meaning the function is zero at the vertex and negative elsewhere, satisfying the statement.
  • If a>0 a > 0 , the parabola opens upward, meaning the function is zero at the vertex and positive elsewhere, contradicting the statement.
  • If a=0 a = 0 , the function is linear and doesn't represent a parabola.

Since no specific information regarding the sign of a a is given, we cannot conclusively state that the function is always negative except at the vertex.

Therefore, the solution is: Cannot be determined.

Answer

Cannot be determined