Radical Functions and Their Graphs

By the end of this lesson, you will be able to graph square root and cube root functions with transformations, identify domain and range, and describe the key features of radical function graphs.

The Big Idea

Radical functions have restricted domains (even roots require non-negative radicands) and produce graphs with characteristic curved shapes. Understanding transformations on radical functions is essential — they are parent functions for a broad family of real-world models, from the period of a pendulum to orbital speeds.

Parent Functions

Square root function: $f(x) = \sqrt{x}$

• Domain: $[0, \infty)$
• Range: $[0, \infty)$
• Starting point: $(0, 0)$
• Shape: increasing, concave down (grows more slowly as $x$ increases)

Cube root function: $g(x) = \sqrt[3]{x}$

• Domain: $(-\infty, \infty)$ (cube roots of negatives are real)
• Range: $(-\infty, \infty)$
• Point of inflection: $(0, 0)$
• Shape: increasing, S-shaped with decreasing growth rate

Transformations

For $f(x) = a\sqrt{b(x-h)} + k$ or $a\sqrt[3]{b(x-h)} + k$:

Parameter	Effect
$h$	Horizontal shift ($+h$ → right, $-h$ → left)
$k$	Vertical shift
$a > 1$	Vertical stretch
$0 < a < 1$	Vertical compression
$a < 0$	Reflection over $x$-axis
$b < 0$	Reflection over $y$-axis

Starting point (square root): the restriction $b(x-h) \geq 0$ determines where the graph starts.

Domain from the Radicand

Square root (even): set radicand $\geq 0$ and solve.

Cube root (odd): all real numbers allowed.

Example 1 — Graphing $f(x) = \sqrt{x + 3} - 2$

Parent: $\sqrt{x}$

Shifts: left 3, down 2

Starting point: $x + 3 = 0 \implies x = -3$ → start at $(-3, -2)$

Domain: $[-3, \infty)$; Range: $[-2, \infty)$