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All courses Algebra II · F-BF.3 26 of 55
Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.

Read signed horizontal and vertical shifts and verify an anchor

Problem
For \(\sqrt{x~-~4}~+~7\), state \(h\) and \(k\), describe the horizontal and vertical shifts, and map the parent endpoint \((0,~0)\).
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Hint

Rewrite x+c as x-(-c), then identify h and k.

In f(x-h)+k, h is the signed horizontal shift and k the signed vertical shift; track an anchor point/asymptote as a check.

Solution walkthrough

01

Match standard shift form

\[\sqrt{x-h}+k=\sqrt{x-4}+7~->~h=4,~k=7\]

The subtraction inside already matches x minus h, while the outside constant is k.

02

Translate the signed parameters

\[h=4~->~\text{right}~4;~k=7~->~\text{up}~7\]

A positive h inside x minus h shifts horizontally right, and positive k shifts vertically up.

03

Map the parent endpoint

\[(x,y)->(x+4,y+7);~(0,0)->(4,7)\]

The square-root parent begins at the origin, so applying both translations places the new endpoint at 4 comma 7.

04

State all requested features

\[\text{Parameters}:~h=4,~k=7.~\text{Shifts}:~\text{right}~4,~\text{up}~7.~\text{Endpoint}:~(0,0)~\text{maps}~\text{to}~(4,7).\]

The mapped anchor verifies both signed shifts, choice A.

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Another way

  1. Set the radicand x minus 4 equal to zero to locate endpoint x equals 4, then evaluate y equals 7.

!

Common mistake

The inside sign looks reversed: x minus 4 means right 4, not left 4.

Solution walkthrough video