Question

Which of the following identifies the quadratic function that is related to the square root function m(x)= square root x-2 +4

Answer 1

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.