Maths with Lemon

Rational Functions

Explore reciprocal graphs, asymptotes, transformations and equations of rational functions for IB Mathematics AA SL.

Introduction to Rational Functions

Prior Knowledge

  • Function notation and domain
  • Solving linear equations
  • Sketching graphs

Key Points

A rational function is a ratio of two polynomials:

\[f(x)=\frac{p(x)}{q(x)},\qquad q(x)\ne 0\]

Start by identifying values that make the denominator zero. These values must be excluded from the domain.

Example

For \(f(x)=\frac{2}{x-3}\), the denominator is zero at \(x=3\).

Domain: \(x\in\mathbb R,\ x\ne3\).

Vertical and Horizontal Asymptotes

What you need to know

  • Identify vertical asymptotes
  • Identify horizontal asymptotes
  • Understand excluded values and holes

Key Points

An asymptote is a line that the graph approaches. For the reciprocal transformation below:

\[f(x)=\frac{a}{x-h}+k,\qquad a\ne0\]
Vertical asymptote
\[x=h\]
Horizontal asymptote
\[y=k\]
Example
\[f(x)=\frac{4}{x+2}-1\]

Vertical asymptote: \(x=-2\). Horizontal asymptote: \(y=-1\).

Be careful: a zero in the original denominator may instead produce a removable hole if a common factor cancels.

Transformations of Rational Functions

What you need to know

  • Horizontal and vertical translations
  • Reflections
  • Vertical stretches and compressions

Key Points

Begin with the parent function:

\[y=\frac1x\]

Its transformed form is:

\[y=\frac{a}{x-h}+k\]
Horizontal movement

\(h\): right if positive, left if negative.

Vertical movement

\(k\): up if positive, down if negative.

The magnitude of \(a\) changes the vertical scale; a negative \(a\) also reflects the graph in the \(x\)-axis relative to the parent graph.

Example
\[y=-\frac{2}{x-3}+1\]

Move right 3 and up 1, reflect in the \(x\)-axis, and stretch vertically by factor 2.

Rational Functions of the Form \(f(x)=\dfrac{ax+b}{cx+d}\)

Learning goals

  • Identify the \(x\)- and \(y\)-intercepts.
  • Find vertical and horizontal asymptotes.
  • Connect the graph to a translated reciprocal function.
  • Recognise exceptional cases.

Key Points

Consider the rational function

\[f(x)=\frac{ax+b}{cx+d},\qquad c\ne0\]

When \(ad-bc\ne0\), its graph is a hyperbola with the same general shape as \(y=\frac1x\), shifted and scaled according to the coefficients \(a,b,c,d\).

\(y\)-intercept

Set \(x=0\), provided \(d\ne0\).

\[\boxed{y=\frac bd}\]
\(x\)-intercept

Set \(y=0\), provided \(a\ne0\) and \(ad-bc\ne0\).

\[\boxed{x=-\frac ba}\]
Vertical asymptote

Set the denominator equal to zero. This is an asymptote when \(ad-bc\ne0\).

\[\boxed{x=-\frac dc}\]
Horizontal asymptote

For large positive or negative \(x\), the leading terms dominate.

\[\frac{ax+b}{cx+d}\longrightarrow\frac ac\]
\[\boxed{y=\frac ac}\]

Equivalently, rewriting the function reveals its reciprocal form:

\[\frac{ax+b}{cx+d}=\frac ac+\frac{bc-ad}{c^2}\cdot\frac{1}{x+\frac dc}\]
Worked example

Find the intercepts and asymptotes of \(f(x)=\dfrac{2x+3}{x-1}\).

\(y\)-intercept
\[f(0)=\frac3{-1}=-3\]
\(x\)-intercept
\[2x+3=0\Rightarrow x=-\frac32\]
Vertical asymptote
\[x=1\]
Horizontal asymptote
\[y=2\]

Indeed, \(f(x)=2+\dfrac5{x-1}\), so the centre of the hyperbola is \((1,2)\).

Special cases: If \(ad-bc=0\), the expression simplifies to a constant for \(x\ne-\frac dc\), with a removable hole rather than a vertical asymptote. If \(d=0\), there is no \(y\)-intercept; if \(a=0\) and \(ad-bc\ne0\), there is no \(x\)-intercept.

Finding Equations from Graphs

Skills

  • Read the two asymptotes
  • Use a point on the curve
  • Check the resulting equation

Key Points

When a graph has the form \(y=\frac{a}{x-h}+k\), first read its asymptotes to determine \(h\) and \(k\).

Worked example

The asymptotes are \(x=2\) and \(y=-1\). The curve passes through \( (4,3) \).

\[y=\frac{a}{x-2}-1\]

Substitute the point:

\[3=\frac{a}{4-2}-1\]
\[4=\frac a2\quad\Rightarrow\quad a=8\]

Equation:

\[\boxed{y=\frac8{x-2}-1}\]

Extra

Material and references

  • Hodder Mathematics AA SL
  • ISBN: 9781510462359
  • Chapter 14C — check against your edition

Key Points

For the function

\[f(x)=\frac{a}{x-h}+k,\quad a\ne0\]
Domain
\[x\ne h\]
Range
\[y\ne k\]

To find an \(x\)-intercept, solve \(f(x)=0\). To find the \(y\)-intercept, substitute \(x=0\), provided it is in the domain.

Exam tip: Label asymptotes and intercepts clearly on any sketch, and use your GDC to check the shape of the graph.

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