IB Maths AA HL · Unit 2: Functions
IB Maths AA HL Transformations and Modulus Questions
Exam-style IB Maths AA HL transformations and modulus questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 20 questions
- Paper 1: 16
- Paper 2: 4
- 7 easy
- 7 medium
- 6 hard
- 3 worked examples
Practise Transformations and Modulus questions →
AA HL formula booklet
What's examined in AA HL transformations and modulus
The question bank covers these transformations and modulus question types (number of questions in brackets):
- Function Graph Transformations (9)
- Modulus Equations & Inequalities (8)
- Modulus Function Properties (3)
Key formulas
- Modulus & argument
- \(r = |z| = \sqrt{a^2 + b^2},\ \theta = \arg z\)
In the same notation as the IB formula booklet. All AA HL formulas →
Transformations and Modulus worked examples
Worked example 1: Applying basic translations · easy
The graph of $f(x) = x^2 - 4$ is translated by the vector $\binom{-2}{3}$ to form the graph of $g(x)$. Find an expression for $g(x)$ in the form $ax^2 + bx + c$.
1. Interpret the translation vector $\binom{-2}{3}$: this shifts the graph $2$ units to the left and $3$ units up.
2. Apply the transformations to the function notation: $g(x) = f(x - (-2)) + 3 = f(x + 2) + 3$.
3. Substitute $(x + 2)$ into the original function: $g(x) = (x + 2)^2 - 4 + 3$.
4. Expand the binomial square: $g(x) = (x^2 + 4x + 4) - 4 + 3$.
5. Simplify the constant terms: $g(x) = x^2 + 4x + 4 - 1$.
6. State the final expanded expression: $\mathbf{g(x) = x^2 + 4x + 3}$.
Examiner tip: Translating left by $c$ units means adding $c$ inside the function bracket (i.e., $f(x+c)$). Reversing this sign and writing $f(x-2)$ for a leftward shift is an incredibly common and costly mistake.
Worked example 2: Solving absolute value equations · medium
Consider the function $f(x) = |2x - 6|$. Solve the equation $f(x) = x$ algebraically.
1. Set up the equation: $|2x - 6| = x$.
2. Form Case 1, assuming the argument is positive (so the modulus does nothing): $2x - 6 = x$.
3. Solve Case 1: Subtract $x$ and add $6$ to get $x = 6$.
4. Form Case 2, assuming the argument is negative (so the modulus multiplies it by $-1$): $-(2x - 6) = x \implies -2x + 6 = x$.
5. Solve Case 2: Add $2x$ to get $3x = 6 \implies x = 2$.
6. Verify both solutions in the original equation to check for extraneous roots. For $x=6$: $|12 - 6| = 6$ (Valid). For $x=2$: $|4 - 6| = |-2| = 2$ (Valid). Solutions: $\mathbf{x = 2, x = 6}$.
Examiner tip: When solving equations with a modulus on one side and a variable on the other, you must check your final answers to ensure they don't produce a negative output for the absolute value, which would be mathematically impossible.
Worked example 3: Advanced composite transformations · hard
The graph of $y = \cos x$ is transformed to the graph of $g(x) = 3\cos(2x - \pi) + 1$. Describe fully the sequence of four geometric transformations required.
1. Factorise the argument of the trigonometric function so the coefficient of $x$ is separated: $2x - \pi = 2(x - \frac{\pi}{2})$.
2. Identify the horizontal stretch from the multiplier $2$: A horizontal stretch by scale factor $\frac{1}{2}$.
3. Identify the horizontal translation from the $(x - \frac{\pi}{2})$ term: A horizontal translation by $\frac{\pi}{2}$ units to the right.
4. Identify the vertical stretch from the multiplier $3$ outside the function: A vertical stretch by scale factor $3$.
5. Identify the vertical translation from the $+1$ at the end: A vertical translation by $1$ unit upwards.
6. State the full sequence. (Note: stretches should generally be applied before translations unless bracketed otherwise).
Examiner tip: Always factorise the coefficient inside the trigonometric argument (i.e. changing $bx-c$ into $b(x - c/b)$) before reading off the horizontal translation. Otherwise, your phase shift will be completely incorrect!
FAQ
How many IB Maths AA HL transformations and modulus questions are there?
There are 20 exam-style transformations and modulus questions in the AA HL question bank (Paper 1: 16 · Paper 2: 4), graded 7 easy, 7 medium, 6 hard. Every question has a full IB-style mark scheme (M, A and R marks).
Is transformations and modulus on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 16 · Paper 2: 4. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
Where can I get the mark schemes?
Open the AA HL Unit 2 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
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