Lesson 3.18

Sketching Rational Graphs

Now we combine everything — VAs, HAs, holes, intercepts, and sign analysis — into a complete, hand-sketched rational function graph.

Introduction

This lesson is the capstone of Unit 3. You'll use every tool from Lessons 3.12–3.17 to produce a complete graph. The key is a systematic checklist — follow it every time and you'll never miss a feature.

Past Knowledge

All of Lessons 3.12–3.17: parent function, VAs, holes, and HA rules.

Today's Goal

Follow a step-by-step checklist to sketch any rational function by hand.

Future Success

Graph analysis skills transfer directly to radical functions (Unit 4) and beyond.

Key Concepts

The Rational Graph Sketching Checklist

1

Factor

Factor numerator and denominator completely.

2

Domain

Find all values where the denominator = 0.

3

Holes

Identify cancelled factors → compute (x, y) of each hole.

4

Vertical Asymptotes

Non-cancelled denominator zeros → dashed vertical lines.

5

Horizontal / Slant Asymptote

Compare degrees → apply the correct rule (3.15, 3.16, or 3.17).

6

x-intercepts

Set the simplified numerator = 0.

7

y-intercept

Evaluate f(0) if 0 is in the domain.

8

Sign chart / test points

Determine which regions are positive or negative.

9

Sketch

Connect the dots, respecting all asymptotes and signs.

Worked Examples

Example 1: Standard Rational Function

Basic

Sketch .

1

VA:

2

HA: deg equal →

3

x-int: →

4

y-int: → same point

Example 2: With a Hole

Intermediate

Sketch .

1

Factor:

2

Hole: cancels → hole at ,

3

VA: | HA: (equal degrees, both coeff = 1)

4

x-int: | y-int:

Example 3: With a Slant Asymptote

Advanced

Sketch .

1

VA: (no cancellation)

2

Slant Asymptote: → SA:

3

x-ints: | No y-int (0 not in domain)

Common Pitfalls

Skipping Steps

Every step matters. Missing the hole check or the y-intercept can produce a completely wrong graph.

Connecting Across a VA

The graph never connects through a vertical asymptote. Each branch is separate.

Real-Life Applications

Engineers and scientists use rational function graphs to model everything from electrical circuits (impedance) to population dynamics. Being able to sketch these by hand gives you an intuitive understanding that software alone can't provide.

Practice Quiz

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