Reciprocal & Rational Functions (DP IB Maths: AA HL)

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Author

Dan

Last updated

15 October 2023

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Reciprocal Functions & Graphs

What is the reciprocal function?

The reciprocal function is defined by $f (x) = \frac{1}{x}, x \neq 0$
Its domain is the set of all real values except 0
Its range is the set of all real values except 0
The reciprocal function has a self-inverse nature
- $f^{- 1} (x) = f (x)$
- $(f \circ f) (x) = x$

What are the key features of the reciprocal graph?

The graph does not have a y-intercept
The graph does not have any roots
The graph has two asymptotes
- A horizontal asymptote at the x-axis: $y = 0$
  - This is the limiting value when the absolute value of x gets very large
- A vertical asymptote at the y-axis: $x = 0$
  - This is the value that causes the denominator to be zero
The graph has two axes of symmetry
- $y = x$
- $y = - x$
The graph does not have any minimum or maximum points

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Linear Rational Functions & Graphs

What is a rational function with linear terms?

A (linear) rational function is of the form $f (x) = \frac{a x + b}{c x + d}, x \neq - \frac{d}{c}$
Its domain is the set of all real values except $- \frac{d}{c}$
Its range is the set of all real values except $\frac{a}{c}$
The reciprocal function is a special case of a rational function

What are the key features of linear rational graphs?

The graph has a y-intercept at $(0, \frac{b}{d})$ provided $d \neq 0$
The graph has one root at $(- \frac{b}{a}, 0)$ provided $a \neq 0$
The graph has two asymptotes
- A horizontal asymptote: $y = \frac{a}{c}$
  - This is the limiting value when the absolute value of x gets very large
- A vertical asymptote: $x = - \frac{d}{c}$
  - This is the value that causes the denominator to be zero
The graph does not have any minimum or maximum points
If you are asked to sketch or draw a rational graph:
- Give the coordinates of any intercepts with the axes
- Give the equations of the asymptotes

Examiner Tip

If you draw a horizontal line anywhere it should only intersect this type of graph once at most
The only horizontal line that should not intersect the graph is the horizontal asymptote
- This can be used to check your sketch in an exam

Worked example

The function $f$ is defined by $f (x) = \frac{10 - 5 x}{x + 2}$ for $x \neq - 2$ .

Write down the equation of

(i)

the vertical asymptote of the graph of

f

(ii)

the horizontal asymptote of the graph of

f

2-4-1-ib-aa-sl-rational-func-a-we-solution

Find the coordinates of the intercepts of the graph of

f

with the axes.

2-4-1-ib-aa-sl-rational-func-b-we-solution

Sketch the graph of

f

2-4-1-ib-aa-sl-rational-func-c-we-solution

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Quadratic Rational Functions & Graphs

How do I sketch the graph of a rational function where the terms are not linear?

A rational function can be written $f (x) = \frac{g (x)}{h (x)}$

Where g and h are polynomials

To find the y-intercept evaluate $\frac{g (0)}{h (0)}$
To find the x-intercept(s) solve $g (x) = 0$
To find the equations of the vertical asymptote(s) solve $h (x) = 0$
There will also be an asymptote determined by what f(x) tends to as x approaches infinity

In this course it will be either:

Horizontal
Oblique (a slanted line)

This can be found by writing $g (x)$ in the form $h (x) Q (x) + r (x)$

You can do this by polynomial division or comparing coefficients

The function then tends to the curve $y = Q (x)$

What are the key features of rational graphs: quadratic over linear?

For the rational function of the form $f (x) = \frac{a x^{2} + b x + c}{d x + e}$
The graph has a y-intercept at $(0, \frac{c}{e})$ provided $e \neq 0$
The graph can have 0, 1 or 2 roots

They are the solutions to $a x^{2} + b x + c = 0$

The graph has one vertical asymptote $x = - \frac{e}{d}$
The graph has an oblique asymptote $y = p x + q$

Which can be found by writing $a x^{2} + b x + c$ in the form $(d x + e) (p x + q) + r$

Where p, q, r are constants
This can be done by polynomial division or comparing coefficients

What are the key features of rational graphs: linear over quadratic?

For the rational function of the form $f (x) = \frac{a x + b}{c x^{2} + d x + e}$
The graph has a y-intercept at $(0, \frac{b}{e})$ provided $e \neq 0$
The graph has one root at $x = - \frac{b}{a}$
The graph has can have 0, 1 or 2 vertical asymptotes

They are the solutions to $c x^{2} + d x + e = 0$

The graph has a horizontal asymptote

Examiner Tip

If you draw a horizontal line anywhere it should only intersect this type of graph twice at most
- This idea can be used to check your graph or help you sketch it

Worked example

The function $f$ is defined by $f (x) = \frac{2 x^{2} + 5 x - 3}{x + 1}$ for $x \neq - 1$ .

(i)

Show that

\frac{2 x^{2} + 5 x - 3}{x + 1} = p x + q + \frac{r}{x + 1}

for constants

p, q

and

r

which are to be found.

(ii)

Hence write down the equation of the oblique asymptote of the graph of

f

2-5-1-ib-aa-hl-quad-rational-function-a-we-solution

Find the coordinates of the intercepts of the graph of

f

with the axes.

2-5-1-ib-aa-hl-quad-rational-function-b-we-solution

Sketch the graph of

f

2-5-1-ib-aa-hl-quad-rational-function-c-we-solution

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Reciprocal & Rational Functions (DP IB Maths: AA HL)

Revision Note

What is the reciprocal function?

What are the key features of the reciprocal graph?

What is a rational function with linear terms?

What are the key features of linear rational graphs?

How do I sketch the graph of a rational function where the terms are not linear?

What are the key features of rational graphs: quadratic over linear?

What are the key features of rational graphs: linear over quadratic?

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

1. Number & Algebra

1.1 Number & Algebra Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

1.1.3 Partial Fractions

1.2 Exponentials & Logs

1.2.1 Introduction to Logarithms

1.2.2 Laws of Logarithms

1.2.3 Solving Exponential Equations

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.3.5 Compound Interest & Depreciation

1.4 Simple Proof & Reasoning

1.4.1 Proof

1.5 Further Proof & Reasoning

1.5.1 Proof by Induction

1.5.2 Proof by Contradiction

1.6 Binomial Theorem

1.6.1 Binomial Theorem

1.6.2 Extension of The Binomial Theorem

1.7 Permutations & Combinations

1.7.1 Counting Principles

1.7.2 Permutations & Combinations

1.8 Complex Numbers

1.8.1 Intro to Complex Numbers

1.8.2 Modulus & Argument

1.8.3 Introduction to Argand Diagrams

1.9 Further Complex Numbers

1.9.1 Geometry of Complex Numbers

1.9.2 Forms of Complex Numbers

1.9.3 Complex Roots of Polynomials

1.9.4 De Moivre's Theorem

1.9.5 Roots of Complex Numbers

1.10 Systems of Linear Equations

1.10.1 Systems of Linear Equations

1.10.2 Algebraic Solutions

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Quadratic Functions & Graphs

2.2.1 Quadratic Functions

2.2.2 Factorising & Completing the Square

2.2.3 Solving Quadratics

2.2.4 Quadratic Inequalities

2.2.5 Discriminants

2.3 Functions Toolkit

2.3.1 Language of Functions

2.3.2 Composite & Inverse Functions

2.3.3 Symmetry of Functions

2.3.4 Graphing Functions

2.4 Other Functions & Graphs

2.4.1 Exponential & Logarithmic Functions

2.4.2 Solving Equations

2.4.3 Modelling with Functions

2.5 Reciprocal & Rational Functions

2.5.1 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.6.1 Translations of Graphs

2.6.2 Reflections of Graphs

2.6.3 Stretches Graphs

2.6.4 Composite Transformations of Graphs

2.7 Polynomial Functions

2.7.1 Factor & Remainder Theorem

2.7.2 Polynomial Division

2.7.3 Polynomial Functions