Lesson 3.12

The Reciprocal Parent Function

The function is the parent function of all rational functions — just as is the parent of all quadratics.

Introduction

Every rational function is a transformation of . Understanding this parent function — its shape, its asymptotes, and its behavior — is the foundation for graphing all rational functions.

Past Knowledge

Domain restrictions (Lesson 3.1) and function transformations from earlier courses.

Today's Goal

Graph and identify its key features: asymptotes, domain, range, and symmetry.

Future Success

Lessons 3.13–3.18 build on this shape to graph more complex rational functions.

Key Concepts

The Graph of

Key Features

Domain: — all reals except 0

Range: — same as domain

Vertical Asymptote: (the y-axis)

Horizontal Asymptote: (the x-axis)

Behavior

Symmetry: Odd function — symmetric about the origin

Quadrants: Lives in Quadrants I and III

As :

As :

Worked Examples

Example 1: Vertical Shift

Basic

Graph and identify the asymptotes.

1

This is shifted up 3

VA stays at , HA moves to

VA: , HA:

Example 2: Horizontal Shift

Intermediate

Graph and identify the asymptotes.

1

This is shifted right 2

VA moves to , HA stays at

VA: , HA:

Example 3: Both Shifts

Advanced

Graph and identify the asymptotes.

1

Shift left 1 and down 2

VA: , HA:

VA: , HA:

Common Pitfalls

Confusing Shift Direction

shifts right 2, not left. The sign in the denominator works like parabola shifts.

Drawing Through the Asymptote

The graph never crosses a vertical asymptote. Each branch stays on its own side.

Real-Life Applications

Inverse-square laws in physics (gravity, light intensity, electric fields) follow the shape of . Understanding this curve helps you predict how quickly forces diminish with distance.

Practice Quiz

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