Lesson 6.7

Graphing Rational Functions

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

Introduction

A Rational Graph isn't just a squiggly line; it's a map defined by its "fences" (asymptotes) and "landmarks" (intercepts and holes). We're going to put everything together to graph the complete function.

Past Knowledge

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

Today's Goal

Combine Holes, Asymptotes, and Intercepts to graph the complete rational function step-by-step.

Future Success

Mastering graphing rational functions is the final step in Precalculus algebra before learning limits in Calculus.

Key Concepts

The 6-Step Strategy

  1. Simplify First: Factor Top/Bottom.
  2. Find Holes: Any cancelled factors? Plot the open circle.
  3. Find VAs: Set remaining denominator to 0. Draw dashed vertical lines.
  4. Find HA/SA: Compare degrees. Draw dashed horizontal/slant line.
  5. Intercepts:
    • x-intercept: Set Top = 0.
    • y-intercept: Set .
  6. 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-Life Applications

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.

Practice Quiz

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