IB Mathematics AA HL More Rational Functions Study Notes | New Syllabus

IB Mathematics AA HL More Rational Functions Study Notes

IB Mathematics AA HL More Rational Functions Study Notes

IB Mathematics AA HL More Rational Functions Study Notes Offer a clear explanation of More Rational Functions , including various formula, rules, exam style questions as example to explain the topics. Worked Out  examples and common problem types provided here will be sufficient to cover for topic More Rational Functions.

Rational Functions

Rational Functions

Rational function is the ratio of two polynomial functions where the denominator polynomial is not equal to zero. It is usually represented as $R(x) =\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomial functions.

 \( f(x) = \frac{ax + b}{cx^2 + dx + e} \)

This type of rational function has a linear numerator and a quadratic denominator.

  • Vertical Asymptotes: Solve \( cx^2 + dx + e = 0 \). The real solutions give vertical asymptotes.
  • Horizontal Asymptote: As \( x \to \infty \), \( f(x) \to 0 \) because degree of denominator (2) > degree of numerator (1).
  • Intercepts:
    • x-intercept: Set numerator = 0 → \( ax + b = 0 \)
    • y-intercept: Set \( x = 0 \)

Example : 

Function: \( f(x) = \frac{2x + 3}{x^2 – x – 2} \) Find all its key featues.

▶️Answer/Explanation
  • Factor denominator: \( x^2 – x – 2 = (x – 2)(x + 1) \)
  • Vertical asymptotes: \( x = 2 \), \( x = -1 \)
  • Horizontal asymptote: \( y = 0 \) (degree of denominator > degree of numerator)
  • Y-intercept: \( f(0) = \frac{3}{-2} = -1.5 \)

The graph approaches the horizontal asymptote \( y=0 \) as \( x \to \pm \infty \).

Second Type of Rational Functions:

\( f(x) = \frac{ax^2 + bx + c}{dx + e} \)

 

This type of rational function has a quadratic numerator and a linear denominator.

  • Vertical Asymptote: Solve \( dx + e = 0 \) → \( x = -\frac{e}{d} \)
  • Oblique (Slant) Asymptote: Since degree of numerator (2) > degree of denominator (1), perform long division. The quotient gives the slant asymptote.
  • Intercepts:
    • x-intercepts: Set numerator = 0 → solve \( ax^2 + bx + c = 0 \)
    • y-intercept: Set \( x = 0 \)

Example : 

Function: \( f(x) = \frac{x^2 + 2x + 1}{x – 1} \) Find all its key featues.

▶️Answer/Explanation
  • Simplify numerator: \( (x + 1)^2 \)
  • Vertical asymptote: \( x = 1 \)
  • Oblique asymptote: Perform division:
    \( \frac{x^2 + 2x + 1}{x – 1} = x + 3 + \frac{4}{x – 1} \)
    So asymptote is \( y = x + 3 \)
  • Y-intercept: \( f(0) = \frac{1}{-1} = -1 \)

The graph approaches the line \( y = x + 3 \) as \( x \to \infty \).

Reciprocal Function

f(x) = 1/x

The reciprocal function is a specific type of rational function:

  

\( f(x) = \frac{1}{x} \)

  • Form: It can be written as \( f(x) = \frac{a}{x} \) where \( a = 1 \).
  • Vertical asymptote: \( x = 0 \)
  • Horizontal asymptote: \( y = 0 \)
  • Key property: The reciprocal function is self-inverse, meaning: \( f^{-1}(x) = f(x) \)
  • Graph: The graph is symmetric in the line \( y = x \) because of the self-inverse property.

The reciprocal function belongs to the family of rational functions where the numerator is a constant and the denominator is linear.

Graphing Rational Function Using Technology

Graphing Rational Functions Using Technology

  • A rational function is of the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.
  • Key features to analyze when graphing rational functions:
    • Vertical asymptotes: Occur at zeros of \( Q(x) \) (denominator).
    • Horizontal or oblique asymptotes: Determined by comparing degrees of \( P(x) \) and \( Q(x) \).
    • Y-intercept: Found by evaluating \( f(0) \) (if defined).
    • X-intercepts: Found by setting \( P(x) = 0 \).
    • End behavior: How the function behaves as \( x \to \infty \) or \( x \to -\infty \).
  • Technology (e.g., Desmos, GeoGebra, GDC) helps plot these functions quickly, identify asymptotes, and verify key points.

Example: 

Function: \( f(x) = \frac{2x + 3}{x^2 – x – 2} \)

▶️Answer/Explanation
  1. Enter the function into graphing technology (Desmos, GeoGebra, GDC).
  2. Find vertical asymptotes: Solve \( x^2 – x – 2 = 0 \):
    \( (x – 2)(x + 1) = 0 \Rightarrow x = 2, -1 \)
  3. Find horizontal asymptote:
    Degree denominator (2) > degree numerator (1): \( y = 0 \)
  4. Find y-intercept: \( f(0) = \frac{3}{-2} = -1.5 \)
  5. Optional: Find x-intercept: Set numerator = 0:
    \( 2x + 3 = 0 \Rightarrow x = -1.5 \)

The graph shows the function approaching \( y=0 \) as \( x \to \pm \infty \), and vertical asymptotes at \( x = -1 \) and \( x = 2 \).

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