Find the Domain of y=-(x+6½)²-2¼: Complete Function Analysis

To solve this problem, we'll determine the domains where the function is positive or negative:

• Given the function $y = -\left(x + 6\frac{1}{2}\right)^2 - 2\frac{1}{4}$, it's in vertex form, with $a = -1$.
• The vertex is at $\left(x, y\right) = \left(-6\frac{1}{2}, -2\frac{1}{4}\right)$, and since $a < 0$, the parabola opens downward.
• Because the parabola opens downward and has no x-intercepts due to $k < 0$, the function $y = 0$ is never zero nor positive.
• This implies there are no values of $x$ for which $y > 0$. Therefore, the positive domain is nonexistent.
• Thus, the negative domain encompasses all $x$ such that $y \leq 0$, which is the entire set of real numbers.

Since the function's value is never positive, the correct description of positive and negative domains is as follows:
Positive domain: None
Negative domain: For all $x$

Therefore, the solution is:

$x < 0 :$ for all $x$

$x > 0 :$ none