Domain Analysis: Finding Valid Inputs for y=-(x-2 6/19)²-2

To solve this problem, we must analyze the quadratic function $y = -\left(x - 2\frac{6}{19}\right)^2 - 2$ to determine its positive and negative domains.

• Step 1: Identify the vertex and direction
  The given function is in the form $y = a(x - h)^2 + k$, where $a = -1$, $h = 2\frac{6}{19}$, and $k = -2$. The vertex of the parabola is at $(2\frac{6}{19}, -2)$.
• Step 2: Analyze the direction of the parabola
  Since $a = -1$ (negative), the parabola opens downward. This indicates the vertex is at the maximum point of the parabola.
• Step 3: Determine the function's values
  Since the maximum value of the function (at the vertex) is $y = -2$, and the parabola opens downward, the function cannot be positive anywhere. It is always less than or equal to $-2$, so it's negative for all $x$.
• Step 4: Establish the positive and negative domains
  Since the function is always negative, there are no positive domains. Therefore, the negative domain for the function is all real numbers x \>.

Therefore, the positive and negative domains are:

\( x > 0 : none

$x < 0 :$ all $x$