Rational Functions in MYP Extended Maths

What Is a Rational Function?

A rational function is a function defined by a ratio of two polynomials. At MYP Extended level, the focus is on the family f(x) = k/x and its variations — functions that produce a characteristic curved shape called a hyperbola. These functions behave very differently from linear and quadratic functions and require careful attention to where they are and are not defined.

The Reciprocal Function: f(x) = 1/x

The basic rational function f(x) = 1/x has two branches, one in the first quadrant and one in the third quadrant. Key features:

• Asymptotes: The graph approaches but never reaches x = 0 (vertical asymptote) and y = 0 (horizontal asymptote).
• No x-intercept and no y-intercept — the function is undefined at x = 0.
• Domain: all real numbers except x = 0.
• Range: all real numbers except y = 0.

Variations: f(x) = k/(x − a) + b

Transformations shift and scale the basic hyperbola:

• Replacing x with (x − a) shifts the vertical asymptote to x = a.
• Adding b shifts the horizontal asymptote to y = b.
• The constant k scales the branches (and a negative k reflects across an axis).

For any rational function in this form, identify the two asymptotes first, then determine which quadrants the branches occupy by testing a point.

Domain Restrictions

Always state the domain restriction explicitly: x ≠ a. This is not optional — a rational function without its domain restriction is an incomplete definition. This links directly to the Domain and Range topic.

Common Mistakes

• Drawing branches that cross the asymptotes — they never do.
• Stating the vertical asymptote as x = 0 for f(x) = 1/(x − 3) — it is x = 3.
• Forgetting to include domain restrictions when defining the function.

MYP Context

Rational functions appear in Criterion A questions (identify features, evaluate) and in Criterion B investigations (how does changing k affect the graph?). Sketch several members of the family f(x) = k/x for different values of k to build intuition.