Question

Which of the following identifies the quadratic function that is related to the square root function m(x)=x−2−−−−−√+4
?(1 point)
Responses

n(x)=4x2+2
n left parenthesis x right parenthesis equals 4 x squared plus 2

n(x)=(x+4)2−2
n left parenthesis x right parenthesis equals left parenthesis x plus 4 right parenthesis squared minus 2

n(x)=(x−4)2+2
n left parenthesis x right parenthesis equals left parenthesis x minus 4 right parenthesis squared plus 2

n(x)=14x2+2

Answer 1

To identify the quadratic function that corresponds to the given square root function \( m(x) = \sqrt{x - 2} + 4 \), we should first understand the relationship between square root functions and quadratic functions.

The function \( m(x) = \sqrt{x - 2} + 4 \) can be rewritten to express \( x \) in terms of \( y \):

1. Subtract 4 from both sides:
\[
m(x) - 4 = \sqrt{x - 2}
\]

2. Square both sides to eliminate the square root:
\[
(m(x) - 4)^2 = x - 2
\]

3. Rearranging gives us:
\[
x = (m(x) - 4)^2 + 2
\]

This indicates that the quadratic function has the form:
\[
n(y) = (y - 4)^2 + 2
\]
where \( y = m(x) \).

Now, we check the provided options to see which one matches this formula:

1. **\( n(x) = 4x^2 + 2 \)**
- This is not a match.

2. **\( n(x) = (x + 4)^2 - 2 \)**
- This is not a match.

3. **\( n(x) = (x - 4)^2 + 2 \)**
- This does match the arrangement we derived, substituting \( y \) for \( x \).

4. **\( n(x) = 14x^2 + 2 \)**
- This is not a match.

Thus, the correct choice that identifies the quadratic function related to the square root function is:
\[
\boxed{n(x) = (x - 4)^2 + 2}
\]