Find the Domain of y=-(x+7)²-5: Negative Quadratic Function Analysis

Find the positive and negative domains of the function below:

$y=-\left(x+7\right)^2-5$

The function given is $y = -\left(x + 7\right)^2 - 5$, which is a parabola that opens downwards. Let's determine the positive and negative domains:

Firstly, we identify the vertex of the function as $(-7, -5)$. The vertex form tells us that the parabola opens downwards because the coefficient of the squared term is negative ($a = -1$). This indicates that the maximum point of the parabola is at the vertex, and the function decreases on either side of the vertex.

Given the downward opening of the parabola and the maximum value $y = -5$ at $x = -7$, the graph of the parabola lies entirely beneath this maximum point. Thus, the function is always non-positive.

Since the function never crosses the x-axis and is below or equal to the vertex's y-coordinate at all points, we find that:

• No values of $x$ give $y > 0$, so the positive domain is empty.
• All real values of $x$ result in $y \leq 0$, indicating a negative domain for $x$.

Therefore, the solutions are:

$x < 0 :$ none

$x > 0 :$ all $x$

The correct choice is:

Choice 4: $x < 0 :$ none

$x > 0 :$ all $x$