Parabola Analysis: Relationship Between Vertex Position and Function Values

To solve this problem, we'll analyze what is implied if the parabola is smiling (opening upwards) and if its vertex is located above the x-axis:

• Step 1: Identify properties of the quadratic function. A "smiling" parabola in the form $y = a(x-h)^2 + k$ with $a > 0$ means the parabola opens upwards. The vertex $(h, k)$ will determine its intersection with the x-axis.
• Step 2: Evaluate the impact of the vertex's location. If the vertex is above the x-axis, then $k > 0$, indicating the vertex point itself is positive.
• Step 3: Assess when the function is negative. For an upwards-facing parabola, $y$ takes on positive values when above the x-axis and negative values when below. If the vertex is above the x-axis, the quadratic will indeed become positive as $x$ moves away from the vertex, indicating that it cannot always be negative.
• Step 4: Interpretation of statement: Given that the parabola could cross the x-axis in some regions other than the vertex and will achieve positive $y$-values whenever the vertex is above the x-axis, the statement "the parabola is always negative" is incorrect.

Therefore, the given statement is Incorrect.