Analysing Rational Functions

Sketch and analyze graphs of rational functions. Grade 12 Math 30-1 Alberta.

Lesson 6.6 of Radical & Rational Functions in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• The Big Discovery — VA vs Point of Discontinuity. Understand why some NPVs create vertical asymptotes while others create holes (points of discontinuity), and master the factoring technique that reveals the difference.
• Full Analysis — Example 1. Work through a complete 9-step analysis of a rational function, from factoring to finding all key features including VA, POD, asymptotes, intercepts, and domain/range.
• Complex Functions, HA Rules & Slant Asymptotes. Master the three rules for horizontal asymptotes, analyze complex factored functions with multiple PODs, and learn when slant asymptotes appear.
• Matching Graphs & Building Equations from Scratch. Learn the four-question strategy to match functions to graphs without a calculator, and discover how to build rational functions from given features like VAs and intercepts.

Inside the lesson — a free preview

Two Functions That Look Similar But Behave Very Differently

Consider These Two Functions. f(x) = (x^2 - 4x + 3)/(x + 1)

g(x) = (x^2 + 4x + 3)/(x + 1)

They look almost identical. Both have (x + 1) in the denominator. Both have NPV x = -1.

But when you graph them, they look completely different. Why?

The answer comes from factoring the numerator.

Step 1 — Always Factor Everything First

For f(x):. Numerator: x^2 - 4x + 3 = (x - 1)(x - 3)

Denominator: (x + 1)

Can anything cancel? Does (x + 1) appear in the numerator? No.

Nothing cancels.

So f(x) = ((x - 1)(x - 3))/(x + 1) — fully simplified as is.

For g(x):. Numerator: x^2 + 4x + 3 = (x + 1)(x + 3)

Denominator: (x + 1)

Can anything cancel? Yes! (x + 1) appears in both top and bottom.

g(x) = ((x + 1)(x + 3))/(x + 1) = (x + 3)

Simplified, with a restriction that x ≠ -1.

This Is the Key Distinction

When Does Each Happen?. When a factor cancels between numerator and denominator, something very different happens compared to when it doesn't cancel:

Factor does NOT cancel → Vertical Asymptote (VA)

The curve shoots off to ±∞ on either side of that x-value. There is a vertical line the curve never crosses.

Factor DOES cancel → Point of Discontinuity (POD)

Also called a "hole." The function simplifies to something nice everywhere except at that one x-value, where there is a single missing point. No asymptote — just a tiny gap.

In Plain English. A VA is a wall the curve can never cross.

A POD is just one missing dot on an otherwise smooth curve.

They look completely different on a graph even though they both come from the same NPV.

Worked example: y = (x^2 + 2x - 8)/(x^2 + 5x + 4)

The Complete Process. This is the complete process you follow every single time you analyze a rational function.

Work through it in order and you won't miss anything.

Create a free Studyio account to take the full lesson with voice-over teaching, practice questions and instant feedback.