To find the quadratic function related to the given square root function \( m(x) = \sqrt{x - 2} + 4 \), we first need to express this function in terms of its vertex.
The square root function \( \sqrt{x - 2} \) has its vertex at the point \( (2, 0) \). The addition of 4 translates this vertex upwards to \( (2, 4) \).
The corresponding quadratic function that has the same vertex can be found by "squaring" the transformation. If we let \( n(x) \) be the quadratic function, we can express it in vertex form as follows:
- Recognize that the vertex is at \( (2, 4) \).
- The general form of a quadratic function in vertex form is given by: \[ n(x) = a(x - h)^2 + k \] where \( (h, k) \) is the vertex.
In this case, \( h = 2 \) and \( k = 4 \): \[ n(x) = a(x - 2)^2 + 4 \]
Since we need to square \( m(x) \) to get \( n(x) \): \[ m(x) = \sqrt{x - 2} + 4 \] Squaring both sides: \[ (m(x) - 4)^2 = x - 2 \] This gives us the corresponding quadratic relationship.
Let's expand and rearrange \( (y - 4)^2 = x - 2 \) to express \( y \) (where \( n(x) \) stands for \( y \)): \[ y - 4 = \sqrt{x - 2} \] Squaring both sides yields: \[ (y - 4)^2 = x - 2 \] Thus, we can rearrange it into standard quadratic form.
From the options provided:
- \( n(x) = 4x^2 + 2 \)
- \( n(x) = (x + 4)^2 - 2 \)
- \( n(x) = (x - 4)^2 + 2 \)
- \( n(x) = \frac{1}{4}x^2 + 2 \)
None of these options directly represent the expression we'd expect from the relation between the square root function and its corresponding quadratic function based solely on transformation. However, using the vertex transformation, the quadratic function we expect should match a form around its vertex.
Since we see \( n(x) = (x - 2)^2 + 4 \) provides the values consistent with the key points (2, 4), upon examining the provided options, it becomes clear that the answers given are not directly relating to the vertex formula we derived.
In conclusion:
To accurately identify the correct relation based on transformations and quadratics, none of the explicit options provided fit. However, if we are to look towards expanding or related forms and if further context indicated directional values to specific forms attempted or common patterns in quadratic shifts were given, it could assist in making precision selection from excess outputs. Please recheck for comprehensiveness on any typographical constraints or restrictions potentially omitted allowing a more accurate frame to identification.