Lesson 4.17

Graphing Cube Root Functions

The cube root function has an elegant "S" shape that passes through the origin. Unlike square roots, it has no domain restriction — cube roots exist for all real numbers.

Introduction

While the square root only lives on the right side of the graph, the cube root stretches in both directions. It's the logical counterpart: odd roots accept negative inputs. The same transformation rules (shift, stretch, reflect) apply.

Past Knowledge

Cube roots (4.5), graphing square root functions (4.16), transformations.

Today's Goal

Graph cube root functions using transformations and compare to square root graphs.

Future Success

Radical function graphs connect to inverse functions — the cube root is the inverse of .

Key Concepts

Parent Function

Domain: Range:

Inflection point:

Square Root vs. Cube Root

Feature
Domain
Shape	Half curve	S-curve
Neg. inputs?	No	Yes

Same transformation form — inflection point at

Worked Examples

Example 1: Shifted

Basic

Graph .

Identify:

Inflection point shifts to

Domain: Range:

Example 2: Reflected

Intermediate

Graph .

Identify:

Reflected over x-axis, shifted left 3. Inflection at

The S-curve is flipped upside down and shifted left

Example 3: Comparing Both Parents

Comparison

Graph and together.

Both pass through (0,0) and (1,1), but the cube root extends to negative x

Common Pitfalls

Restricting the Domain

Unlike , the cube root has no domain restriction. Don't write for cube root functions.

Confusing Starting Point with Inflection

Square roots have a starting point (endpoint). Cube roots have an inflection point (the curve passes through, not starts at).

Real-Life Applications

Cube root functions model situations where growth slows symmetrically in both directions — such as the relationship between volume and side length of a cube, or wind chill calculations in meteorology.