To identify the quadratic function related to the square root function given by \( m(x) = \sqrt{x - 2} + 4 \), we can first express the square root function in terms of its inverse, which is a quadratic function.
The function \( m(x) = \sqrt{x - 2} + 4 \) can be rewritten as:
\[ m(x) - 4 = \sqrt{x - 2} \]
Now, if we square both sides to eliminate the square root, we get:
\[ (m(x) - 4)^2 = x - 2 \]
Next, we can rearrange this equation to express \( x \) in terms of \( m(x) \):
\[ x = (m(x) - 4)^2 + 2 \]
Now, let's introduce \( y = m(x) \) for clarity. We then have:
\[ x = (y - 4)^2 + 2 \]
This represents a quadratic function in terms of \( y \):
\[ x = (y - 4)^2 + 2 \]
This indicates that the quadratic function related to the square root function \( m(x) = \sqrt{x - 2} + 4 \) can be expressed as:
\[ x = y^2 - 8y + 16 + 2 \]
Simplifying this gives:
\[ x = y^2 - 8y + 18 \]
Thus, the quadratic function related to the given square root function is:
\[ y^2 - 8y + (18 - x) = 0 \]
This means, while the root function itself does not represent a classic quadratic function, its relationship can be expressed through the derived equation above. The standard form of the quadratic equation can indeed be written as:
\[ x = (y - 4)^2 + 2 \]
This establishes the relationship between the square root function and its corresponding quadratic function.
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