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All courses Math III · F-BF.3 19 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
Write sqrt(x-4)+7 as f(x-h)+k relative to its parent. Enter signed h, horizontal direction/distance, signed k, vertical direction/distance, and the shifted anchor feature.
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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

Start from the parent function

\[\text{parent}:~y~=~\text{sqrt}(x)\]

The graph is a transformation of the square-root parent function, so compare each change to \(y=sqrt(x)\).

02

Read the horizontal change

\[x~->~x-4~\text{means}~\text{right}~4\]

Replacing \(x\) by \(x-4\) means the graph needs an input \(4\) units larger to match each parent input, so it shifts right.

03

Read the vertical change

\[+7~\text{means}~\text{up}~7\]

Adding \(7\) outside the square root increases every output by \(7\), so the graph moves upward.

04

Check a key point

\[(0,0)~->~(4,7)\]

The starting point of \(sqrt(x)\) moves from \((0,0)\) to \((4,7)\), confirming right \(4\) and up \(7\).

05

State the final answer

\[\text{parent}~f(x)=\text{sqrt}(x);~\text{standard}~\text{form}=f(x-h)+k;~\text{signed}~h=4;~\text{horizontal}~\text{shift}=\text{right}~4;~\text{signed}~k=7;~\text{vertical}~\text{shift}=\text{up}~7;~\text{anchor}~\text{feature}=\text{endpoint}~\text{moves}~(0,0)→(4,7)\]

The question asks for both shifts, so state the horizontal shift and the vertical shift together.

+

Another way

  1. You can also track the starting point of \(sqrt(x)\): it moves from \((0,0)\) to \((4,7)\), which confirms the same horizontal and vertical shifts.

!

Common mistake

A common mistake is to say \(x-4\) means left \(4\), but inside-function horizontal shifts go in the opposite direction.