Find the Domain of (x-12)²+4: Analyzing Positive and Negative Ranges

To find the positive and negative domains of the quadratic function $y = (x - 12)^2 + 4$, let's proceed step-by-step:

• Step 1: Identify the structure.
  The function is in vertex form $y = (x - 12)^2 + 4$, which indicates a parabola that opens upwards, with vertex $(12, 4)$.
• Step 2: Determine the minimum value.
  Since the vertex form shows the minimum value at $y = 4$ when $x = 12$, the function never actually reaches negative values.
• Step 3: Analyze positivity.
  Given that the minimum value $y = 4$ when $x = 12$, and the parabola opens upwards, every possible value of $x$ results in $y \geq 4$. Therefore, the function is always positive for all $x$.
• Step 4: Conclusion on domains.
  The function has no negative values for any input. Thus, the negative domain is none, and the positive domain includes all values of $x$. Therefore, we assert the positive domain is: all $x$.

With our analysis complete, we can conclude that the positive and negative domains of the function are:

$x < 0 :$ none

$x > 0 :$ all $x$