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All courses Algebra I · F-IF.7.b 68 of 76
Graph square-root, cube-root, absolute-value, step, and piecewise-defined functions, showing key features by hand in simple cases and with technology when the functions are more complicated.

Transform a fixed set of square-root parent points

Problem
For \(y=\sqrt{x}\), evaluate the parent inputs \(u=0,1,4,9\) and record each point \((u,\sqrt{u})\). Use the first point and the square-root restrictions to state the endpoint, domain, and range, then identify the point-and-feature set supported by your calculations.
Source graph of the square-root parent points (0,0), (1,1), (4,2), and (9,3), corresponding to u=0,1,4,9. Open full size
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Hint

Write the coordinate rule before calculating the four points.

Let u be the radicand. Transform (u,√u), not a random selection of x-values.

Solution walkthrough

01

Choose inputs with exact square roots

\[u=0,1,4,9\]

These nonnegative parent inputs are perfect squares, so their square-root outputs can be evaluated exactly.

02

Evaluate every parent point

\[\begin{aligned} \sqrt{0}=0~->~(0,0) \\ \sqrt{1}=1~->~(1,1) \\ \sqrt{4}=2~->~(4,2) \\ \sqrt{9}=3~->~(9,3) \end{aligned}\]

Each ordered pair uses the chosen input first and its principal, nonnegative square root second.

03

Identify the endpoint and domain

\[\begin{aligned} \text{first}~\text{point}~(0,0) \\ \text{radicand}~x\ge~0~->~\text{domain}~[0,\text{infinity}) \end{aligned}\]

The real square-root function begins where its radicand is zero. Negative real inputs are not allowed, so (0,0) is its endpoint.

04

Determine the range and check the plot

\[\sqrt{x}\ge~0~->~\text{range}~[0,\text{infinity})\]

Principal square roots never produce negative outputs. The inspected graph begins at the origin and extends rightward and upward through all four points.

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Another way

  1. Start from output values 0,1,2,3 and square them to obtain the corresponding inputs 0,1,4,9.

!

Common mistake

Do not include negative outputs such as -2 for sqrt(4). The square-root function uses the principal nonnegative root.

Answer graph of y=√x using u=0,1,4,9: points (0,0), (1,1), (4,2), and (9,3), endpoint (0,0), domain [0,∞), and range [0,∞).
rule=(u,√u)→(u,√u); u-values=0,1,4,9; points=(0,0),(1,1),(4,2),(9,3); endpoint=(0,0); domain=[0,∞); range=[0,∞); checks=√0=0,√1=1,√4=2,√9=3

Solution walkthrough video