Find the Domain of (x+15)²+6: Analyzing Valid Input Values

To determine the positive and negative domains of the function $y=(x+15)^2+6$, we start by analyzing its structure.

The function is given in vertex form, $y=a(x-h)^2+k$, where $a=1$, $h=-15$, and $k=6$. Since $a=1 > 0$, the parabola opens upwards.

1. Vertex and Axis of Symmetry:
- Vertex: The vertex of the parabola is at $(-15, 6)$. This indicates the minimum point since the parabola opens upwards.

2. Range of the function:
- As $(x+15)^2$ is always zero or positive, the smallest value for $y$ is when $(x+15)^2=0$, thus $y=6$. Hence, $y \geq 6$.

3. Analyzing the function's values:
- Since the minimum value of $y$ is 6 and it increases as $x$ moves away from -15 in either direction, the function does not achieve any negative values.

4. Conclusion:
- The function is always positive, $y \geq 6$.

Based on this analysis:

Negative domain: The function does not have any negative values, thus, for $x < 0$, there are no values where the function is negative.

Positive domain: The entire domain is positive. Therefore, for $x > 0$, the function remains positive for all $x$.

Thus, the positive and negative domains are:

$x < 0 :$ None

$x > 0 :$ All $x$