UpStudy Free Solution:

To determine the function that results from applying the given transformations to the graph of \(y = \sqrt { x} \), we will apply each transformation step-by-step.

Shift up 7 units:
Shifting the graph of \(y = \sqrt { x} \) up by 7 units results in:
\[y = \sqrt { x} + 7\]

Reflect about the \(x\)-axis:
Reflecting the graph \(y = \sqrt { x} + 7\) about the \(x\)-axis changes the sign of \(y\):
\[y = - ( \sqrt { x} + 7) = - \sqrt { x} - 7\]

Reflect about the \(y\)-axis:
Reflecting the graph \(y = - \sqrt { x} - 7\) about the \(y\)-axis changes the sign of \(x\):
\[y = - \sqrt { - x} - 7\]
Thus, the function that is finally graphed after all the transformations is:
\[y = - \sqrt { - x} - 7\]
 

Supplemental Knowledge

When transforming the graph of a function, each type of transformation affects the function in a specific way:

Vertical Shifts: Shifting a graph up or down involves adding or subtracting a constant to the function. For example, shifting \(y = f( x) \) up by \(k\) units results in \(y = f( x) + k\).

Reflections:

Reflecting about the \(x\)-axis changes \(y = f( x) \) to \(y = - f( x) \).

Reflecting about the \(y\)-axis changes \(y = f( x) \) to \(y = f( - x) \).
Let's apply these transformations step-by-step to the function \(y = \sqrt { x} \):

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