Rewrite functions in equivalent forms to reveal and explain useful properties.
Identify root graph features directly
Problem
Analyze \(y~=~\sqrt{x~-~5}~+~2\). State the root type and radicand condition; give the domain, range, and endpoint or center; describe the transformation and reflection; and state the monotonic direction.
Big Picture
What this problem is really about
First identify the root family, since a square root has a one-sided endpoint and requires a nonnegative radicand. Setting the radicand to zero locates that endpoint, while the inside and outside constants describe its horizontal and vertical shifts. The outside coefficient controls scale and reflection, which in turn determine the range and whether the parent’s increasing direction is preserved.
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