In algebraic modeling, functions can take many forms depending on how variables interact. When one polynomial expression is divided by another, it forms a rational function. These functions exhibit fascinating graphical properties, such as sudden breaks, structural boundaries, and curves known as hyperbolas. Mastering rational functions is a core component of GCSE Higher extension and A-Level Mathematics, providing vital skills for analyzing algebraic fractions and graph theory.

The best Maths tutors available

Theory

What is a Rational Function?

A rational function is defined mathematically as the quotient of two polynomial functions. It can be written in the general form:

f(x) = \dfrac{P(x)}{Q(x)}

Where P(x) and Q(x) are polynomials, and the denominator Q(x) ≠ 0. If Q(x) is a constant, such as 1, the rational function simplifies into a standard polynomial. Therefore, all polynomial functions are technically a subcategory of rational functions.

Domain Restrictions

The foundational rule of mathematics dictates that division by zero is undefined. Consequently, a rational function cannot accept any input value for x that causes its denominator to equal zero. The domain of a rational function consists of all real numbers except for the specific roots of the denominator equation Q(x) = 0.

For example, consider the base reciprocal function:

y = \dfrac{1}{x}

If x = 0, the output climbs to infinity, creating a critical break in the graph. Thus, the domain restriction is x ≠ 0.

Asymptotes of a Hyperbola

The characteristic graph of a simple rational function is a hyperbola. As the graph approaches an undefined point, the curve bends sharply, tracking alongside an imaginary boundary line known as an asymptote. The curve gets infinitely close to this line but never touches or crosses it. There are three primary types of asymptotes:

Vertical Asymptotes: Occur at the x-values where the denominator is zero (Q(x) = 0).
Horizontal Asymptotes: Show the behavior of the graph as x approaches positive or negative infinity.
Oblique (Slanted) Asymptotes: Occur when the degree of the numerator polynomial is exactly one higher than the degree of the denominator polynomial.

Asymptopes of a hyperbola - oblique, vertical and horizontal graph — Image Source: Gianpiero Placidi

Transformations of Hyperbolas

The base hyperbola:

y = \dfrac{1}{x}

has its center at the origin (0, 0), where its vertical asymptote (x = 0) and horizontal asymptote (y = 0) intersect. We can shift this curve across the coordinate plane using standard geometric transformations.

1. Vertical Translation

Adding or subtracting a constant outside the fraction shifts the graph vertically up or down. The transformed equation is:

y = \dfrac{k}{x} + a

If a > 0, the curve and its horizontal asymptote shift upwards by "a" units. The new center of the hyperbola becomes (0, a).

2. Horizontal Translation

Modifying the input variable within the denominator shifts the graph horizontally left or right. The transformed equation is:

y = \dfrac{k}{x + b}

If b > 0, the curve shifts to the left by "b" units because the undefined point now occurs where x + b = 0, meaning x = -b. The new center of the hyperbola is (-b, 0).

Exam Tip: Watch out for the sign in the denominator. A positive sign shifts the graph in the negative x-direction, while a negative sign shifts it in the positive x-direction.

3. Oblique (Combined) Translation

When horizontal and vertical translations occur simultaneously, the hyperbola undergoes a combined shift parallel to both axes. The general transformed equation is written as:

y = a + \dfrac{k}{x + b}

Under this condition, the vertical asymptote shifts to x = -b and the horizontal asymptote shifts to y = a. The intersection of these asymptotes, which defines the center of the hyperbola, moves to the coordinate point (-b, a).

Graphs of the different transformations of hyperbolas - vertical, horizontal and oblique with a reference base graph — Image Source: Gianpiero Placidi

Worked Example

Problem: Analyze the rational function given by the equation:

y = \dfrac{3}{x - 2} + 5

Determine the domain restrictions, state the equations of both asymptotes, and find the coordinates of the center of the hyperbola.

Step-by-step Solution:

Find the vertical asymptote: Set the denominator equal to zero to discover where the function breaks. x - 2 = 0 implies x = 2 Therefore, the vertical asymptote is the line x = 2.
Determine the domain: Since the function is undefined at the vertical asymptote, the domain is all real numbers except 2. Domain:

x \in \mathbb{R}, x \neq 2

Find the horizontal asymptote: Look at the constant added outside the fraction block. As x approaches infinity, the fraction term approaches 0, leaving y approaching 5. Therefore, the horizontal asymptote is the line y = 5.
Identify the center point: The center of the transformed hyperbola is located at the intersection point of the two asymptotes (-b, a). Center = (2, 5)

Practice Questions & Solutions

Score:

State the equations of the vertical and horizontal asymptotes for the rational function:

$\text{[math]}$

Solution

To find the vertical asymptote, set the denominator expression equal to zero:

$\text{[math]}$

Solving for x gives the vertical asymptote line:

$\text{[math]}$

The horizontal asymptote is given by the constant vertical displacement term outside the fraction block:

$\text{[math]}$

Identify the domain restriction for the following rational function:

$\text{[math]}$

Solution

The function becomes undefined when the denominator expression evaluates to zero:

$\text{[math]}$

Isolate the variable term to find the excluded x-value:

$\text{[math]}$

Therefore, the domain of the function is restricted such that:

$\text{[math]}$

Find the coordinates of the center of the hyperbola defined by the equation:

$\text{[math]}$

Solution

Identify the horizontal shift from the denominator to find the x-coordinate of the center:

$\text{[math]}$

Identify the vertical shift from the constant term to find the y-coordinate of the center:

$\text{[math]}$

Combine these values to state the coordinate pairs for the center of the hyperbola:

$\text{[math]}$

A hyperbola has its center located at the coordinate point below:

$\text{[math]}$

If the scale factor k equals 1, construct its rational equation.

Solution

The general structural form for a translated hyperbola is given by:

$\text{[math]}$

The center coordinate maps directly to the tracking parameters:

$\text{[math]}$

This reveals the values for our constants:

$\text{[math]}$

Substitute these parameters along with k equals 1 into the general form:

$\text{[math]}$

An advanced linear rational function is written as:

$\text{[math]}$

Convert this via algebraic division into the form below to determine its horizontal asymptote:

$\text{[math]}$

Solution

Rewrite the numerator expression so that it contains a matching multiple of the denominator:

$\text{[math]}$

Simplify the constant terms in the adjusted numerator:

$\text{[math]}$

Split the rational expression into two separate fraction components:

$\text{[math]}$

Cancel out the common linear binomial bracket from the first term:

$\text{[math]}$

This reveals that as x approaches infinity, the horizontal asymptote is:

$\text{[math]}$

Summarise with AI:

Did you like this article? Rate it!

5.00 (3 rating(s))

Gianpiero Placidi

UK-based Chemistry graduate with a passion for education, providing clear explanations and thoughtful guidance to inspire student success.

Formulas

Sequence Formulas | Arithmetic, Geometric and More

Theory
Worked Example
Practice Questions & Solutions

Understanding Rational Functions and their Hyperbolas

Theory

What is a Rational Function?

Domain Restrictions

Asymptotes of a Hyperbola

Transformations of Hyperbolas

1. Vertical Translation

2. Horizontal Translation

3. Oblique (Combined) Translation

Worked Example

Practice Questions & Solutions

Theory

Exponent Rules

Composition of Functions

Constant Functions

Convergent and Divergent Sequences

Even and Odd Functions

Exponential Function

Graphing Parabolas

Arithmetic Sequence

Mean Value Theorem

Rolle’s Theorem

Intervals of Increase and Decrease

Periodic Functions

Graphing rational function I

Graphing rational function III

Inflection Points

Increasing and Decreasing Functions

Piecewise Functions

Trigonometric Functions

Linear Function

Domain of a Function

Maxima, Minima and Inflection Points

Linear Functions Worksheet

Monotone Sequence

Radical Functions

Function

Geometric Sequence

x- and y-Intercepts

Bounded Sequence: Monotonic and Non-Monotic

L’Hôpital’s Rule

Graphing polynomial function I

Absolute and Relative Maxima and Minima

Maxima and Minima of Functions

Types of Functions

Asymptote Functions

Coordinates in the Plane

Optimization

nth Term

Graphing rational function V

Limit of a Sequence

Inverse Function

Quadratic Function

Absolute Value Function

Sequences

Properties of Limits

Geometric Sequence Problems And Solutions

Arithmetic Sequence Problems with Solutions

Logarithmic Functions

Concave and Convex Functions: Theory and Examples

Rational Functions: Graphs, Hyperbolas and Asymptotes Explained

Bounded Function Examples: Theory and Practice Questions

Formulas

Sequence Formulas | Arithmetic, Geometric and More

Exercises

Inflection Points Worksheet

Sequences Problems

Concavity and Convexity Worksheet

Increase and Decrease Worksheet

Limit of Sequence Problems

Applications of Derivatives Worksheet

Inverse Functions Worksheet

Exponential, Logarithmic and Trigonometric Functions Worksheet

Composition of Functions Worksheet

Graphing Worksheet

Mean Value Theorem Word Problems

Maximum and Minimum Word Problems

Quadratic Function Word Problems

Rational Functions Worksheet

Domain Worksheet