Problem

Derive the exact graph fingerprint for \(y=-\sqrt{x+2}+1\). Identify \(\text{both}~\text{translations}\) and the reflection, locate the endpoint, determine direction, domain, and range, and verify the fingerprint by evaluating \(x=-1\) and \(x=2\).

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Square-root graph shifted left 2 and up 1 without reflection; endpoint (−2,1); increasing; domain [−2,∞); range [1,∞); points (−1,2), (2,3).

Square-root graph shifted left 2 and down 1, then reflected; endpoint (−2,−1); decreasing; domain [−2,∞); range (−∞,−1]; points (−1,−2), (2,−3).

Square-root graph shifted right 2 and up 1, then reflected; endpoint (2,1); decreasing; domain [2,∞); range (−∞,1]; points (3,0), (6,−1).

Square-root graph shifted left 2 and up 1, then reflected; endpoint (−2,1); decreasing; domain [−2,∞); range (−∞,1]; points (−1,0), (2,−1).

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