Find Domains of y=-(x-9)²-3: Analyzing Positive and Negative Regions

To solve this problem, let's first examine the given quadratic function:

$y = -\left(x - 9\right)^2 - 3$

This function is in vertex form $y = a(x - h)^2 + k$, where:

• $a = -1$
• $h = 9$
• $k = -3$

From the values of $a$, $h$, and $k$:

• The vertex of the parabola is $(9, -3)$.
• Since $a < 0$, the parabola opens downwards.

Next, we investigate the function's behavior to determine its positive and negative values:

• The vertex $(9, -3)$ is the maximum point of the parabola because it opens downwards. It implies that the highest value $y$ attains is $-3$.
• Given the vertex form, the entire curve will lie below this maximum or be equal to it, hence the function never attains positive values. Therefore, there are no positive values for $y$.
• For the negative domain, the function's value will always be less than or equal to $-3$ (i.e., negative).

Thus, we can conclude:

• There is $x$ such that $y$ is positive.
• All values of $x$ will make the function assume negative values, specifically for all real numbers $x$.

Therefore, the solution to the problem is:

$x < 0 :$ none

$x > 0 :$ all $x$