Lesson 4.16

Graphing Square Root Functions

The graph of has a distinctive "eyebrow" shape — it starts at a point and curves gently upward. Transformations shift, stretch, and reflect this shape.

Introduction

The parent function has domain and range . It starts at the origin and increases slowly. All transformations follow the same rules as other parent functions.

Past Knowledge

Square roots (4.5), function transformations, domain and range.

Today's Goal

Graph square root functions using transformations and identify domain/range.

Future Success

Cube root graphing (4.17) uses the same transformation ideas with a different parent shape.

Key Concepts

Parent Function

Domain: Range:

Starting point:

Transformation Form

h = horizontal shift (right if +)

k = vertical shift (up if +)

a = vertical stretch/reflect

Starting point =

Worked Examples

Example 1: Shifted Right and Up

Basic

Graph and state domain/range.

Identify:

Starting point:

Domain: Range:

Example 2: Reflected

Intermediate

Graph .

Identify:

Reflected over x-axis, shifted left 1, up 4. Starting point:

Domain: Range:

Example 3: Vertical Stretch

Advanced

Compare , , and .

stretches (steeper), compresses (flatter)

Common Pitfalls

Direction of Horizontal Shift

shifts right 2 (not left). The sign is opposite inside the function.

Forgetting the Domain Changes

The domain shifts with the starting point. has domain , not .

Real-Life Applications

The square root function models diminishing returns — like how the first few hours of studying improve your score dramatically, but additional hours yield progressively less improvement.