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International Baccalaureate Mathematics

Functions

Syllabus Content

The reciprocal function f(x)=1/x, x ≠ 0: its graph and self-inverse nature.
Rational functions of the form f(x)=(ax+b)/(cx+d) and their graphs.
Equations of vertical and horizontal asymptotes

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Furthermore

Official Guidance, clarification and syllabus links:

Sketches should include all horizontal and vertical asymptotes and any intercepts with the axes.

Link to: transformations (SL2.11).

Vertical asymptote: \(x=-\frac{d}{c}\)

Horizontal asymptote: \(y=\frac{a}{c}\)

Rational functions of the form \( f(x) = \frac{ax + b}{cx + d} \), where \( a, b, c, \) and \( d \) are constants and \( cx + d \neq 0 \), exhibit interesting behaviors on their graphs. These functions can have both vertical and horizontal asymptotes, which are lines that the graph approaches but never touches.

Vertical asymptotes occur where the denominator of the function is zero. For the given form, the vertical asymptote is at \( x = -\frac{d}{c} \), provided \( c \neq 0 \). Horizontal asymptotes, on the other hand, are determined by the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the x-axis, or \( y = 0 \). If the degrees are equal, the horizontal asymptote is \( y = \frac{a}{c} \).

Key Formulae:
- Vertical Asymptote: \( x = -\frac{d}{c} \) (if \( c \neq 0 \))
- Horizontal Asymptote: \( y = 0 \) if degree of numerator < degree of denominator; \( y = \frac{a}{c} \) if degrees are equal

Example:
Consider the function \( f(x) = \frac{2x + 3}{x - 1} \). The vertical asymptote is at \( x = 1 \) (since \( x - 1 = 0 \) at \( x = 1 \)), and the horizontal asymptote is \( y = \frac{2}{1} = 2 \) since the degrees of the numerator and denominator are equal.

The graph of this function approaches but does not touch the lines \( x = 1 \) and \( y = 2 \), illustrating the properties of its asymptotes.

If you use a TI-Nspire GDC there are instructions for finding asymptotes.

This video on Rational Functions, Asymptotes and Graphs is from Revision Village and is aimed at students taking the IB Maths Standard level course.

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