Lesson 6.7

Graphing Rational Functions

The ultimate boss battle of Chapter 6. Combining Holes, Asymptotes, and Intercepts into one complete graph.

Introduction

Prerequisite Connection: We have spent weeks studying the individual pieces: Domain (6.2), VAs (6.3), HAs (6.4), Holes (6.5), and Intercepts.

Today's Increment: We are now putting it all together. A Rational Graph isn't just a squiggly line; it's a map defined by its "fences" (asymptotes) and "landmarks" (intercepts and holes).

Explanation of Key Concepts

The 6-Step Strategy

Simplify First: Factor Top/Bottom.
Find Holes: Any cancelled factors? Plot the open circle.
Find VAs: Set remaining denominator to 0. Draw dashed vertical lines.
Find HA/SA: Compare degrees. Draw dashed horizontal/slant line.
Intercepts:
- x-intercept: Set Top = 0.
- y-intercept: Set .
Test Points: Pick x-values in the empty zones to see if the curve is Above or Below the asymptote.

Worked Examples

Level: Basic

Example 1: The Full Analysis

Graph .

Step 1: Simplify

. Nothing cancels. No holes.

Step 2: Asymptotes

VA:

HA: Degrees equal (1 vs 1). Ratio:

Step 3: Intercepts

x-int: Top=0

. Point (2,0).

y-int: Set x=0

. Point (0,-4).

Level: Intermediate

Example 2: Hidden Simplification

Graph .

Factor:

cancels! Use simplified form

for drawing.

The Catch: Draw the line

, but erase the point at

.
Hole Location: Plug 3 into simplified:

. Hole at (3,6).

Level: Advanced

Example 3: Volcano Graph

Sketch .

VA:

. Multiplicity 2 (squared), so "Volcano" (same direction).
Sign Check: Top is always negative (-2). Bottom is always positive (squared). Result is Negative.

Both branches go down to

. It looks like a waterfall.

Common Pitfalls

Forgetting to graph the Hole:
If you don't draw the open circle, you have graphed the wrong function (the simplified one, not the original).
Crossing the VA:
Never cross a Vertical Asymptote. It is undefined terrain. You CAN cross a Horizontal Asymptote in the middle, but never a Vertical one.

Real-World Application

Physics: Resistance

In a parallel circuit with variable resistance , the total resistance might be .

The graph passes through (0,0) (no resistance means short circuit). As , the graph approaches (Horizontal Asymptote). You can never get more than 10 ohms out of this parallel setup, no matter how big the resistor is.