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)=(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
n left parenthesis x right parenthesis equals Start Fraction 1 over 4 End Fraction x squared plus 2

n(x)=4x2+2

Answer 1

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:
\[
\boxed{n(x) = (x - 4)^2 + 2}
\]