Find the Domain of y=-(x-9)²+4: Analyzing Quadratic Functions

To solve the problem, we'll find the roots of the quadratic function:

• Step 1: Set $y = 0$ in the equation $-\left(x - 9\right)^2 + 4 = 0$.
• Step 2: Solve for $x$:
  $-\left(x - 9\right)^2 + 4 = 0$ leads to $\left(x - 9\right)^2 = 4$.
  Take the square root: $x - 9 = \pm 2$.
• Step 3: Solve for $x$:
  $x - 9 = 2$ gives $x = 11$.
  $x - 9 = -2$ gives $x = 7$.
• Step 4: Analyze intervals determined by $x = 7$ and $x = 11$:
• For $x < 7$, choose a value (e.g., $x = 0$): $y = -\left(0 - 9\right)^2 + 4 = -81 + 4 = -77$ (negative).
• For $7 < x < 11$, choose a value (e.g., $x = 9$): $y = -\left(9 - 9\right)^2 + 4 = 4$ (positive).
• For $x > 11$, choose a value (e.g., $x = 12$): $y = -\left(12 - 9\right)^2 + 4 = -9 + 4 = -5$ (negative).

Therefore, the positive and negative domains of the function are:

$x > 11$ or $x < 0 : x < 7$

$x > 0 : 7 < x < 11$

In summary, the correct intervals where the function is positive or negative are identified. The function is positive for $7 < x < 11$ and negative otherwise.