Find the Domain of y=-(x+4)²-1: Complete Analysis

Find the positive and negative domains of the function below:

$y=-\left(x+4\right)^2-1$

To solve this problem, we must determine the intervals where the function $y = -\left(x + 4\right)^2 - 1$ is positive or negative.

Let's follow these steps:

• Step 1: Identify the vertex of the quadratic function. The function is given in vertex form, $-\left(x + 4\right)^2 - 1$, where the vertex is $(-4, -1)$.
• Step 2: Determine the direction of the parabola. Since the coefficient of the squared term is negative (i.e., $-1$), the parabola opens downwards.
• Step 3: Analyze the graph. A downward-opening parabola means that the function reaches its maximum at the vertex, and thus all other points on the parabola are below this maximum value of $-1$.
• Step 4: Determine the sign of $y$. Because the vertex $y$-value is $-1$ and all other points are below it, the function does not take on any positive values. Therefore, the positive domain of $y$ is nonexistent, and the function is entirely in the negative $y$-domain.

Therefore, the positive domain is empty because the parabola of $y = -\left(x + 4\right)^2 - 1$ does not reach any positive $y$-values. Thus, the function is negative for all $x$.

In conclusion, the correct answer is: $x < 0 :$ none and $x > 0 :$ all $x$.