All courses Math III · F-IF.7.b 24 of 55
Graph square-root, cube-root, absolute-value, step, and piecewise-defined functions.

Graph a transformed square root from exact anchor rows

Problem
For y=sqrt(x-2)+1, enter a,h,k, endpoint, domain, range, reflection/vertical-scale fields, and one transformed anchor row for each parent radicand 0,1,4,9.
Parent square-root graph with standard anchor dots on a coordinate grid. Open full size
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Hint

Identify a,h,k and solve x-h=r for each selected radicand r.

For y=a√(x-h)+k, endpoint is (h,k); use radicands 0,1,4,9 for easy exact anchors.

Solution walkthrough

01

Identify the parent graph

\[y~=~\sqrt{x}\]

The parent square-root graph starts at \((0,0)\) and increases to the right.

02

Find the horizontal and vertical shifts

\[y~=~\sqrt{x-2}+1\]

The \(x-2\) shifts the graph right 2 units, and the \(+1\) shifts it up 1 unit.

03

Locate the endpoint, domain, and range

\[\text{endpoint}~(2,1),~\text{domain}~x~\ge~2,~\text{range}~y~\ge~1\]

The graph now begins at \((2,1)\), the domain is \(x \ge 2\), and the range is \(y \ge 1\).

04

Use an anchor point and state the features

\[\text{standard}~\text{parameters}=a=1,h=2,k=1;~\text{endpoint}=(2,1);~\text{domain}=[2,∞);~\text{range}=[1,∞);~x-\text{axis}~\text{reflection}=\text{no};~\text{vertical}~\text{scale}=1;~\text{anchor}~\text{rows}=(\text{radicand}~0→(2,1));(1→(3,2));(4→(6,3));(9→(11,4))\]

A nearby anchor point helps sketch the shape after the shift: when \(x=3\), the output is \(2\).

+

Another way

  1. You can substitute values that make the radicand a perfect square, such as \(x=2\) and \(x=3\), to build the graph from easy points.

!

Common mistake

A common mistake is to put the endpoint at \((-2,1)\), but \(x-2\) shifts the graph right, not left.

Square-root graph shifted to endpoint 2 comma 1 and passing through 3 comma 2.
Endpoint (2, 1); anchor (3, 2); domain x ≥ 2; range y ≥ 1.