IB Mathematics AA – Topic 2: Functions

Comprehensive Guide to Rational Functions

Introduction to Rational Functions

A rational function is a function that can be expressed as the quotient (ratio) of two polynomials. These functions exhibit fascinating behavior including asymptotes (lines the graph approaches but never touches) and discontinuities, making them essential for modeling real-world phenomena like rates of change, concentration problems, and electrical circuits.

General form: \(f(x) = \frac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) \neq 0\). The function is undefined wherever the denominator equals zero, creating vertical asymptotes or holes.

Key features: Unlike polynomials, rational functions can have vertical asymptotes (where the function "blows up"), horizontal or oblique asymptotes (end behavior), and distinctive hyperbolic shapes. Understanding these features is crucial for accurate graphing and analysis.

In this guide: We'll explore how to find vertical, horizontal, and oblique asymptotes, locate intercepts, understand reciprocal functions (the simplest rational functions), master sketching techniques, and effectively use your GDC to verify and analyze these functions for IB exams.

1. Asymptotes

What is an Asymptote?

An asymptote is a line that a function's graph approaches arbitrarily closely but never actually reaches (or crosses, in most cases). Asymptotes reveal the function's behavior at extreme values and where it's undefined.

Three Types of Asymptotes

Type 1: Vertical Asymptotes

Definition: Vertical line \(x = a\) where function approaches \(\pm\infty\)

How to find: Set denominator equal to zero and solve

For \(f(x) = \frac{P(x)}{Q(x)}\): Solve \(Q(x) = 0\)

Important: Check if common factors cancel (creates a hole, not an asymptote!)

Example: \(f(x) = \frac{1}{x-3}\) has vertical asymptote at \(x = 3\)

Type 2: Horizontal Asymptotes

Definition: Horizontal line \(y = b\) that function approaches as \(x \to \pm\infty\)

How to find: Compare degrees of numerator and denominator

Case 1: Degree of \(P(x)\) < Degree of \(Q(x)\) → \(y = 0\)

Case 2: Degree of \(P(x)\) = Degree of \(Q(x)\) → \(y = \frac{\text{leading coeff of }P}{\text{leading coeff of }Q}\)

Case 3: Degree of \(P(x)\) > Degree of \(Q(x)\) → No horizontal asymptote (may have oblique)

Example: \(f(x) = \frac{2x^2+1}{x^2-4}\) has horizontal asymptote at \(y = 2\)

Type 3: Oblique (Slant) Asymptotes

Definition: Diagonal line \(y = mx + b\) that function approaches as \(x \to \pm\infty\)

When it occurs: Degree of numerator = Degree of denominator + 1

How to find: Perform polynomial long division

\(\frac{P(x)}{Q(x)} = mx + b + \frac{R(x)}{Q(x)}\)

Oblique asymptote: \(y = mx + b\)

Example: \(f(x) = \frac{x^2+1}{x}\) has oblique asymptote \(y = x\)

⚠ Common Pitfalls:

Holes vs. asymptotes: If \((x-a)\) cancels from both numerator and denominator, it's a hole at \(x = a\), not an asymptote
Function can cross horizontal asymptote: Unlike vertical asymptotes, graphs can cross horizontal ones
Forgetting to simplify first: Always factor and cancel common terms before finding asymptotes
Wrong horizontal asymptote: Must compare degrees correctly

2. Finding Intercepts

x-Intercepts and y-Intercepts

Finding Intercepts:

y-Intercept

Method: Set \(x = 0\) and evaluate \(f(0)\)

\(f(0) = \frac{P(0)}{Q(0)}\)

Coordinates: \((0, f(0))\)

Note: Only one y-intercept possible (if \(Q(0) \neq 0\))

Example: For \(f(x) = \frac{x+1}{x-2}\), y-intercept is \((0, -\frac{1}{2})\)

x-Intercepts (Roots/Zeros)

Method: Set \(f(x) = 0\) and solve

\(\frac{P(x)}{Q(x)} = 0 \Rightarrow P(x) = 0\)

Key insight: Fraction equals zero when numerator = 0 (and denominator ≠ 0)

Coordinates: \((a, 0)\) where \(a\) is a root of \(P(x) = 0\)

Note: Can have 0, 1, or multiple x-intercepts

Example: For \(f(x) = \frac{x^2-4}{x+1}\), x-intercepts are \((2, 0)\) and \((-2, 0)\)

💡 Intercept Tips:

x-intercepts found from numerator only (set numerator = 0)
If numerator has no real roots, function has no x-intercepts
y-intercept doesn't exist if \(x = 0\) is a vertical asymptote
Use GDC to verify intercepts after calculating algebraically

Example 1: Finding Asymptotes and Intercepts

Problem: For \(f(x) = \frac{3x - 6}{x + 2}\), find:

(a) All asymptotes

(b) All intercepts

Solution:

(a) Asymptotes:

Vertical asymptote: Set denominator = 0

\(x + 2 = 0\)

\(x = -2\)

Vertical asymptote: \(x = -2\)

Horizontal asymptote: Compare degrees

Numerator degree: 1, Denominator degree: 1 (equal)

Horizontal asymptote = \(\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}\)

\(= \frac{3}{1} = 3\)

Horizontal asymptote: \(y = 3\)

(b) Intercepts:

y-intercept: Set \(x = 0\)

\(f(0) = \frac{3(0) - 6}{0 + 2} = \frac{-6}{2} = -3\)

y-intercept: \((0, -3)\)

x-intercept: Set numerator = 0

\(3x - 6 = 0\)

\(3x = 6\)

\(x = 2\)

x-intercept: \((2, 0)\)

3. Reciprocal Functions

The Parent Reciprocal Function

The simplest rational function is the reciprocal function \(f(x) = \frac{1}{x}\). Understanding its properties helps analyze more complex rational functions.

Properties of \(f(x) = \frac{1}{x}\):

Domain: \(x \in \mathbb{R} \setminus \{0\}\) (all real numbers except 0)
Range: \(y \in \mathbb{R} \setminus \{0\}\) (all real numbers except 0)
Vertical asymptote: \(x = 0\) (y-axis)
Horizontal asymptote: \(y = 0\) (x-axis)
Intercepts: None (function never crosses either axis)
Symmetry: Odd function (symmetric about origin): \(f(-x) = -f(x)\)
Shape: Two branches (hyperbola) in quadrants I and III

Transformations of Reciprocal Functions

General Form:

\(f(x) = \frac{a}{x - h} + k\)

\(h\): Horizontal shift (vertical asymptote moves to \(x = h\))
\(k\): Vertical shift (horizontal asymptote moves to \(y = k\))
\(a\): Vertical stretch/compression and reflection
- If \(a > 0\): branches in quadrants I and III (relative to asymptotes)
- If \(a < 0\): branches in quadrants II and IV (reflected)

For \(f(x) = \frac{ax + b}{cx + d}\) (Linear/Linear):

This can be rewritten as \(f(x) = \frac{a}{c} + \frac{bc - ad}{c(cx + d)}\)

Vertical asymptote: \(x = -\frac{d}{c}\)
Horizontal asymptote: \(y = \frac{a}{c}\)
Shape: Transformed reciprocal (hyperbola)

Example 2: Reciprocal Function Transformations

Problem: Consider \(f(x) = \frac{2}{x + 3} - 1\)

(a) Identify all transformations from the parent function \(y = \frac{1}{x}\)

(b) State the asymptotes

Solution:

(a) Transformations from \(y = \frac{1}{x}\):

Vertical stretch by factor of 2 (coefficient 2)
Horizontal shift left 3 units (\(x + 3\) in denominator)
Vertical shift down 1 unit (\(-1\) outside)

(b) Asymptotes:

Vertical: Denominator = 0

\(x + 3 = 0 \Rightarrow x = -3\)

Vertical asymptote: \(x = -3\)

Horizontal: Value approached as \(x \to \pm\infty\)

As \(x \to \pm\infty\), \(\frac{2}{x+3} \to 0\), so \(f(x) \to -1\)

Horizontal asymptote: \(y = -1\)

(c) Domain and Range:

Domain: all \(x\) except vertical asymptote

Domain: \(x \in \mathbb{R} \setminus \{-3\}\)

Range: all \(y\) except horizontal asymptote

Range: \(y \in \mathbb{R} \setminus \{-1\}\)

4. Sketching Rational Functions

Step-by-Step Sketching Process

Systematic Approach:

Simplify the function (factor and cancel common terms)
Find vertical asymptotes (set denominator = 0)
Find horizontal/oblique asymptotes (compare degrees or use division)
Find intercepts (x: set numerator = 0; y: evaluate \(f(0)\))
Identify any holes (canceled factors)
Plot asymptotes as dashed lines
Plot intercepts and holes
Test points in each region (between asymptotes)
Sketch smooth curves approaching asymptotes
Verify with GDC

Behavior Near Asymptotes

Understanding Function Behavior:

Near vertical asymptote \(x = a\):
- Test values slightly less than \(a\): does \(f(x) \to +\infty\) or \(-\infty\)?
- Test values slightly more than \(a\): does \(f(x) \to +\infty\) or \(-\infty\)?
Near horizontal asymptote \(y = b\):
- Function approaches from above or below
- May cross the horizontal asymptote (unlike vertical)
In each region between asymptotes:
- Function is continuous (no breaks)
- Smooth curve connecting key points

⚠ Sketching Mistakes:

Crossing vertical asymptotes: Function NEVER crosses vertical asymptotes
Wrong curve shape: Must approach asymptotes correctly (not perpendicular)
Missing regions: Sketch in ALL regions separated by vertical asymptotes
Not labeling: Always label asymptotes and key points clearly

Example 3: Complete Sketch (IB-Style)

Problem: Sketch \(f(x) = \frac{x^2 - 1}{x - 2}\), showing all asymptotes, intercepts, and key features.

Solution:

Step 1: Simplify

\(f(x) = \frac{(x-1)(x+1)}{x-2}\)

No common factors to cancel

Step 2: Vertical asymptote

Set denominator = 0: \(x - 2 = 0\)

Vertical asymptote: \(x = 2\)

Step 3: Oblique asymptote (degree numerator > degree denominator)

Perform long division: \(\frac{x^2-1}{x-2}\)

\(x^2 - 1 = (x-2)(x+2) + 3\)

So: \(f(x) = x + 2 + \frac{3}{x-2}\)

Oblique asymptote: \(y = x + 2\)

Step 4: Intercepts

x-intercepts: Set numerator = 0

\(x^2 - 1 = 0 \Rightarrow x = \pm 1\)

x-intercepts: \((-1, 0)\) and \((1, 0)\)

y-intercept: \(f(0) = \frac{0-1}{0-2} = \frac{-1}{-2} = \frac{1}{2}\)

y-intercept: \((0, \frac{1}{2})\)

Step 5: Test behavior near \(x = 2\)

Just left of 2 (e.g., \(x = 1.9\)): \(f(1.9) = \frac{2.61}{-0.1} < 0\) → goes to \(-\infty\)

Just right of 2 (e.g., \(x = 2.1\)): \(f(2.1) = \frac{3.41}{0.1} > 0\) → goes to \(+\infty\)

Sketch features:

Vertical asymptote \(x = 2\) (dashed vertical line)
Oblique asymptote \(y = x + 2\) (dashed diagonal line)
x-intercepts at \((-1, 0)\) and \((1, 0)\)
y-intercept at \((0, 0.5)\)
Left of \(x = 2\): curve through intercepts, approaching asymptotes
Right of \(x = 2\): curve from \(+\infty\), approaching oblique asymptote

Use GDC to verify sketch is correct

5. Using Your GDC (Graphing Calculator)

Essential GDC Techniques

Key Calculator Functions:

1. Graphing Rational Functions

Enter function in Y= menu (use parentheses carefully!)
Adjust window to see asymptotes clearly
Look for gaps/jumps at vertical asymptotes
Note: GDC may connect across asymptote—ignore false connections

2. Finding Key Features

Zeros (x-intercepts): Use CALC → zero function
y-intercept: Use VALUE function at \(x = 0\)
Vertical asymptotes: Look for undefined values in TABLE
Behavior: Trace along curve to see values

3. Verifying Asymptotes

Vertical: Set TABLE to show x-values near suspected asymptote, observe function "blow up"
Horizontal: Check very large positive and negative x-values (use TABLE)
Oblique: Graph both function and line, observe they get closer at extremes

💡 GDC Pro Tips:

Always show algebraic work first, then verify with GDC
Use DOT mode (not CONNECTED) to avoid false lines at asymptotes
Zoom out to see full behavior and all asymptotes
Use TABLE feature to find exact values at specific points
Screenshot/sketch from GDC for your answer where appropriate

📋 Rational Functions Quick Reference

Feature	How to Find	Notes
Vertical Asymptote	Set denominator = 0	Check for cancellation first
Horizontal Asymptote	Compare degrees of P and Q	Can cross this asymptote
Oblique Asymptote	Long division when deg(P) = deg(Q)+1	Diagonal line
x-intercept	Set numerator = 0	May have 0, 1, or many
y-intercept	Evaluate \(f(0)\)	At most one
Hole	Find canceled factors	Removable discontinuity

🎯 IB Exam Strategy

Common Question Types:

"Find all asymptotes": Check vertical, horizontal, and oblique systematically
"Sketch the function": Find asymptotes and intercepts first, then curve shape
"State domain/range": Exclude vertical asymptote values (domain) and horizontal asymptote value (range)
"Find intercepts": x: numerator = 0; y: evaluate at 0

Time-Saving Tips:

Use GDC to verify your algebraic work quickly
Know degree comparison rules for horizontal asymptotes by heart
Practice polynomial long division for oblique asymptotes
Always simplify/factor before finding asymptotes

🎉 Master Rational Functions!

Rational functions are powerful tools for modeling real-world relationships. Understanding asymptotes, intercepts, reciprocal transformations, and sketching techniques gives you complete control over these functions—essential for IB success and applications in physics, engineering, and economics!

Key Success Factors:

✓ Vertical asymptotes: denominator = 0 (check for cancellation!)
✓ Horizontal asymptotes: compare degrees of numerator and denominator
✓ x-intercepts from numerator; y-intercept from \(f(0)\)
✓ Reciprocal parent function: \(f(x) = \frac{1}{x}\) is foundation
✓ Always sketch asymptotes as dashed lines
✓ Use GDC to verify but show algebraic work first

Simplify First • Find Asymptotes • Sketch Systematically

Master rational functions and excel in IB Mathematics! 🚀