DP IB Maths: AA HL

Revision Notes

IBMaths: AA HLDPRevision Notes2. Functions2.9 Further Functions & Graphs2.9.2 Modulus Transformations

2.9.2 Modulus Transformations

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 anything

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

Exam 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

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Dan

Author: Dan

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.