Graphing Rational Functions: Complete Sketch

Lesson 2 of 2

In this lesson:

• Sign analysis: direction of approach at each asymptote
• Seven-step sketching procedure for rational functions

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Apply sign analysis to determine graph direction in each region
2. Sketch a rational function using all key features
3. Verify hand sketches with technology
4. Explain why functions can cross horizontal asymptotes

You Have the Features — Now Connect Them

From Lesson 1, you can find:

• Zeros: where the graph crosses the x-axis
• Holes: single missing points
• Vertical asymptotes: where the graph blows up
• Horizontal asymptotes: where the graph levels off

Missing piece: which way does the curve approach each asymptote?

Your Complete Seven-Step Sketching Process

1. Factor numerator and denominator
2. Find zeros (check denominator ≠ 0 there)
3. Identify holes from shared factors
4. Find VAs from uncanceled denominator zeros
5. Find HA by comparing degrees
6. Sign analysis: one test point per region
7. Sketch: draw each region's curve

Sign Analysis Fills In the Graph

Vertical asymptotes and zeros create regions on the x-axis.

In each region: one test point determines the sign.

• Positive → graph above x-axis
• Negative → graph below x-axis
• Near a VA: sign tells which direction the curve approaches

Complete Example: Features Identified First

• Zero at ; no holes
• VAs at ,
• HA at (degree 1 < degree 2)

Complete Example: Applying Sign Analysis

Sign tells direction: above x-axis if positive, below if negative

Complete Example: The Finished Sketch

Surprise: Functions Can Cross Their HA

The function has HA at .

Does it cross the x-axis? Set :

At , — the function crosses its own horizontal asymptote!

Use Technology to Check Your Work

After hand sketching, graph in Desmos and compare:

• VA locations and count
• Zero locations and count
• End behavior approaching HA
• Curve direction near each VA

Guided Practice: Analyze This Function

Set up the complete analysis:

1. Zero? Holes? VAs? HA?
2. Create a sign table
3. Sketch the graph

Work through all seven steps before checking the answer.

Guided Practice: Full Solution Revealed

• Zero at ; VAs at , ; HA at
• Sign table: positive, negative, positive, negative
• Sketch: 4 pieces approaching VAs from correct directions

Verify with Desmos.

Sketch These Rational Functions Independently

For each: find zeros, holes, VAs, HA, sign table, sketch.

Practice Answers: All Features Identified

1. : zero at ; VAs at , ; HA:

2. : zeros at ; VAs at ; HA: ; no holes

Lesson 2 Summary: Complete Sketching

✓ Seven steps: factor → zeros → holes → VAs → HA → sign → sketch

✓ Sign analysis: one test point per region

✓ Functions CAN cross their HA for finite x-values

Watch out: HA is a limit at infinity — not a barrier

Complete Toolkit: Rational Function Graphing

You can now:

• Find all zeros, holes, VAs, HAs from a factored expression
• Apply sign analysis to determine graph direction
• Produce a complete hand sketch and verify with technology

Next: transformations of functions and function comparisons