# Rational Functions | Maths with Lemon

Source: https://mathswithlemon.io/aa-functions-rational-functions.html


### 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:

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

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:

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:

Its transformed form is:

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

\(k\): 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.

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


#### 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 \(k\).

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

Substitute the point:

Equation:


#### Extra


##### Material and references

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

##### Key Points

For the function

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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