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Algebra 2 Radical functions

Graphing radical functions

20 practice questions 0 video lessons Theory + worked examples

Graphing Radical Functions

Texas Algebra II (TEKS) • Standard 2A.2(A) • Radical Functions

Graphing Radical Functions is a topic in Radical Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.2(A), which requires students to graph the square root and cube root parent functions and analyze their key attributes.

The square root graph starts at a point with domain \(x\ge0\); the cube root graph is defined for all real numbers.

Texas Algebra II (TEKS) › Radical Functions › Graphing Radical Functions  —  Standard 2A.2(A)

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Theory

Radical parent functions come in two kinds:
  • Square root \(\sqrt{x}\): domain \([0,\infty)\), starts at the origin.
  • Cube root \(\sqrt[3]{x}\): domain all reals, passes through the origin.

They shift with \(\sqrt{x-h}+k\), moving the start point to \((h,k)\).

Even roots need a non-negative radicand; odd roots accept any real number.
Transforming a square root graph The graph of the square root shifts right 2 and up 1, moving its start point. x y √x √(x-2)+1 start (2,1)
\(\sqrt{x-2}+1\) starts at \((2,1)\).
The cube root graph The cube root is defined for all real numbers and increases through the origin. x y ∛x
The cube root is defined for all reals.

Radical parents:

\[f(x)=\sqrt{x-h}+k,\qquad f(x)=\sqrt[3]{x-h}+k\]
the square root and cube root parents, shifted by h and k
Set the radicand \(\ge 0\) to find a square-root domain.

How to graph a radical

  1. Find the domain (radicand \(\ge0\) for even roots).
  2. Locate the start point \((h,k)\).
  3. Plot a few points to the right.
  4. Draw the increasing curve.
Example 1 — Domain of a square root
Find the domain of \(f(x)=\sqrt{x-3}\).
Solution

The radicand must be \(\ge 0\).

\(x-3\)\(\ge\)\(0\)
\(x\)\(\ge\)\(3\)
the domain is x at least 3
Example 2 — Describe the transformation
Describe \(f(x)=\sqrt{x-2}+1\).
Solution

Shift the parent right \(2\), up \(1\).

\(\text{start point}\)\(=\)\((2,1)\)
shifted right 2 and up 1, starting at 2 comma 1
Example 3 — Domain and range of √x
Give the domain and range of \(f(x)=\sqrt{x}\).
Solution

Both are non-negative.

\(\text{domain}\)\(=\)\([0,\infty)\)
\(\text{range}\)\(=\)\([0,\infty)\)
domain and range both 0 to infinity
Example 4 — Cube root domain
What is the domain of \(f(x)=\sqrt[3]{x}\)?
Solution

An odd root accepts negatives too.

\(\text{domain}\)\(=\)\((-\infty,\infty)\)
the cube root's domain is all real numbers

Common pitfalls

Even-root domains are restricted; odd roots are not.
The start point is \((h,k)\) for \(\sqrt{x-h}+k\).
\((x-h)\) shifts right, as always.

Frequently asked questions

What is the domain of \(\sqrt{x}\)?

\([0,\infty)\) — the radicand must be non-negative.

What is the domain of \(\sqrt[3]{x}\)?

All real numbers.

Where does \(\sqrt{x-2}+1\) start?

At the point \((2,1)\).

Why do square roots have restricted domains?

You can't take an even root of a negative number in the reals.

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