Modulus Transformations

How do I sketch the graph of the modulus of a function: y = |f(x)|?

STEP 1: Keep the parts of the graph of y = f(x) that are on or above the x-axis
STEP 2: Any parts of the graph below the x-axis get reflected in the x-axis

How do I sketch the graph of a function of a modulus: y = f(|x|)?

STEP 1: Keep the graph of y = f(x) only for x ≥ 0
STEP 2: Reflect this in the y-axis

What is the difference between y = |f(x)| and y = f(|x|)?

The graph of y = |f(x)| never goes below the x-axis

It does not have to have any lines of symmetry

The graph of y = f(|x|) is always symmetrical about the x-axis

It can go below the x-axis

When multiple transformations are involved how do I determine the order?

The transformations outside the function follow the same order as the order of operations
- $y = |a f (x) + b|$
  - Deal with the a then the b then the modulus
- $y = a |f (x)| + b$
  - Deal with the modulus then the a then the b
The transformations inside the function are in the reverse order to the order of operations
- $y = f (|a x + b|)$
  - Deal with the modulus then the b then the a
- $y = f (a |x| + b)$
  - Deal with the b then the a then the modulus

Examiner Tip

When sketching one of these transformations in an exam question make sure that the graphs do not look smooth at the points where the original graph have been reflected
- For $y = |f (x)|$ the graph should look "sharp" at the points where it has been reflected on the x-axis
- For $y = f (|x|)$ the graph should look "sharp" at the point where it has been reflected on the y-axis

Worked example

The diagram below shows the graph of $y = f (x)$ .

2-9-1-we-diagram

(a) Sketch the graph of $y = |f (x)|$ .

2-9-1-ib-aa-hl-modulus-trans-a-we-solution

(b) Sketch the graph of $y = f (|x|)$ .

2-9-1-ib-aa-hl-modulus-trans-b-we-solution

Modulus Transformations (DP IB Maths: AA HL)

Revision Note

How do I sketch the graph of the modulus of a function: y = |f(x)|?

How do I sketch the graph of a function of a modulus: y = f(|x|)?

What is the difference between y = |f(x)| and y = f(|x|)?

When multiple transformations are involved how do I determine the order?

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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

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1.10 Systems of Linear Equations

1.10.1 Systems of Linear Equations

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2.1.1 Equations of a Straight Line

2.2 Quadratic Functions & Graphs

2.2.1 Quadratic Functions

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2.2.3 Solving Quadratics

2.2.4 Quadratic Inequalities

2.2.5 Discriminants

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2.4.2 Solving Equations

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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

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2.6.4 Composite Transformations of Graphs

2.7 Polynomial Functions

2.7.1 Factor & Remainder Theorem

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2.8 Inequalities

2.8.1 Solving Inequalities Graphically