Find the Domain of y=-(x+7)²: Analyzing Negative Quadratic Functions

To solve this problem, we'll investigate the function $y = -(x+7)^2$.

• Step 1: Recognize that the function $y = -(x+7)^2$ is a quadratic function opening downwards due to the negative coefficient.
• Step 2: The vertex form $y = a(x-h)^2 + k$ helps identify that the vertex of this quadratic is $(-7, 0)$.
• Step 3: As $(x+7)^2 \geq 0$ for any real number $x$, $-(x+7)^2 \leq 0$ is always true. Thus, the function is never positive.
• Step 4: Identify when $y = 0$: This occurs only when $x = -7$.
• Step 5: The domain of $x$ where $y < 0$ occurs when $x \ne -7$, covering all other $x$ in the real numbers since $y$ is negative.

Considering the choices provided, the statement matches choice 3:

$x < 0 : x\ne-7$

$x > 0 :$ none

Therefore, the function has no positive domain values, while the negative domain consists of all values except $x = -7$.