Find the Domain of y=-(x-14)²-6: Positive and Negative Analysis

Find the positive and negative domains of the function below:

$y=-\left(x-14\right)^2-6$

The function given is $y = -\left(x-14\right)^2-6$.

This is a quadratic function in vertex form: $y = a(x-h)^2 + k$ where $a = -1$, $h = 14$, and $k = -6$. The vertex of the function is at $(14, -6)$ and since $a = -1$, the parabola opens downwards.

Step 1: Identify intervals for negative and positive values:
- The vertex at $(14, -6)$ is the maximum point of the parabola.
- For the quadratic to have positive values, $y$ must be greater than 0. Given the vertex and opening direction of the parabola, there are no $x$ values for which $y$ is positive because the parabola is entirely below the x-axis.

Step 2: Analyze $y$ values when $x > 0$ and $x < 0$:
- The parabola is below the x-axis ($y < 0$) for all $x$. Therefore, when checking for $x > 0$, the function remains negative for all positive $x$.

Conclusion: This shows that the function is not positive for any $x$, but is negative for all $x$.

Therefore, the positive and negative domains are as followed:

• $x < 0 :$ none
• $x > 0 :$ all $x$

The correct answer is Choice 2.