To find the quadratic function \( n(x) \) that is related to the square root function \( m(x) = \sqrt{x - 2} + 4 \), we need to recognize the relationship between the square root and squared functions. Specifically, if \( y = \sqrt{x - 2} + 4 \), we can square both sides to solve for \( x \).
Starting with: \[ y - 4 = \sqrt{x - 2} \] Square both sides: \[ (y - 4)^2 = x - 2 \] Rearranging gives us: \[ x = (y - 4)^2 + 2 \]
Now, we can express \( n(x) \) in terms of \( y \): \[ n(y) = (y - 4)^2 + 2 \]
To transform this into the standard form of a quadratic function, we can rewrite and expand it: \[ n(y) = (y - 4)^2 + 2 \] \[ = (y^2 - 8y + 16) + 2 \] \[ = y^2 - 8y + 18 \]
Now, our goal is to express it in \( n(x) \). Since \( y \) is just a placeholder variable, we can replace it with \( x \): Thus: \[ n(x) = (x - 4)^2 + 2 \]
From the options provided:
- n(x) = (x - 4)² + 2
This option matches our derived expression.
The other options do not match:
- n(x) = (x + 4)² − 2
- n(x) = (1/4)x² + 2
- n(x) = 4x² + 2
Thus, the answer is: \[ \boxed{n(x) = (x - 4)^2 + 2} \]