IB Mathematics AA SL The rational function Study Notes

Rational Functions

The study of reciprocal and rational functions is essential for understanding complex relationships in algebra and graphing. These functions are foundational in various fields such as calculus, physics, and economics.

1. Reciprocal Function

The reciprocal function is defined as:

\( f(x) = \frac{1}{x} \), where \( x \neq 0 \)

Properties:

• Self-Inverse Nature: The reciprocal function is its own inverse. Applying the function twice returns the original value: \( f(f(x)) = x \).
• Graph: The graph of \( f(x) = \frac{1}{x} \) consists of two branches, located in the first and third quadrants. It is a hyperbola with asymptotes.

2. Rational Functions

A rational function is a ratio of two linear expressions, given by:

\( f(x) = \frac{ax + b}{cx + d} \)

where \( a, b, c, \) and \( d \) are constants, and \( cx + d \neq 0 \) to avoid division by zero.

Graphical Features:

Vertical Asymptotes: The vertical asymptote occurs where the denominator is zero, i.e., \( cx + d = 0 \). Solving for \( x \) gives:

\( x = -\frac{d}{c} \)

Horizontal Asymptotes: The horizontal asymptote is determined by the ratio of the leading coefficients as \( x \) approaches infinity:

\( y = \frac{a}{c} \)

3. Example Problem

Sketch the graph of \( f(x) = \frac{2x – 3}{x + 4} \).

• Vertical Asymptote: Set \( x + 4 = 0 \), so \( x = -4 \).
• Horizontal Asymptote: As \( x \to \infty \), the horizontal asymptote is \( y = \frac{2}{1} = 2 \).
• Intercepts:
  • x-intercept: Set \( f(x) = 0 \), so \( 2x – 3 = 0 \); solving gives \( x = 1.5 \).
  • y-intercept: Set \( x = 0 \); solving gives \( y = \frac{-3}{4} = -0.75 \).

Graph Analysis: The graph approaches \( x = -4 \) as a vertical asymptote and \( y = 2 \) as a horizontal asymptote. It passes through the points \( (1.5, 0) \) and \( (0, -0.75) \).

4. Connections to Transformations

The graph of a rational function can be transformed similarly to other functions. For instance, translating \( f(x) = \frac{1}{x} \) by \( h \) units horizontally and \( k \) units vertically results in:

\( f(x) = \frac{1}{x – h} + k \)

This transformation affects the position of the asymptotes and the overall shape of the graph.

5. Historical Development

The concept of functions and graphing has been developed by notable mathematicians such as:

• René Descartes (France): Known for Cartesian coordinates and the use of algebra to represent geometric shapes.
• Gottfried Wilhelm Leibniz (Germany): Co-founder of calculus, contributed significantly to the notation and analysis of functions.
• Leonhard Euler (Switzerland): Made profound contributions to graph theory, analysis, and introduced the concept of a function as a fundamental part of mathematics.

6. Use of Technology

Dynamic graphing software, such as Desmos or GeoGebra, can be used to explore rational functions interactively. Using sliders to adjust \( a, b, c, \) and \( d \) helps visualize the effect on the graph’s shape and position of asymptotes.

7.Enrichment: Deriving Horizontal Asymptote Formula

To derive the horizontal asymptote for \( f(x) = \frac{ax + b}{cx + d} \), consider the limit as \( x \to \infty \):

\( \lim_{x \to \infty} \frac{ax + b}{cx + d} = \frac{a}{c} \)

This approach shows why the horizontal asymptote is \( y = \frac{a}{c} \).