Find the Domain of y=(x+2)²+12: Positive and Negative Regions

Find the positive and negative domains of the function below:

$y=\left(x+2\right)^2+12$

To find the positive and negative domains of the function $y = (x+2)^2 + 12$, follow these steps:

• Identify the vertex of the function: The vertex form is $y = (x+2)^2 + 12$, hence the vertex is at $(-2, 12)$.
• Determine the parabola's direction: Given the coefficient of $(x+2)^2$ is positive, the parabola opens upwards.
• Consider the vertex's role: At $x = -2$, the minimum value of $y$ is 12. Since the parabola opens upwards from there, $y \geq 12$ for all $x$.
• Analyze positivity/negativity: Since the minimum $y$-value is 12, the function is always positive for all real $x$, and hence it is not negative for any $x$.

Therefore, the positive domain is all $x$, and there is no negative domain. The final choice is:

$x < 0 :$ none

$x > 0 :$ all $x$