Topic 2.16 · HL only

Graphs with modulus functions: notes and practice questions

Q: Is Graphs with modulus functions SL or HL?

Graphs with modulus functions is HL only. SL students are not examined on it.

Q: How do I revise Graphs with modulus functions for IB Maths AA?

Start from the core idea: the absolute value function |x| is defined as x for x ≥ 0 and -x for x < 0, resulting in a V-shaped graph with its vertex at the origin. In the exam: usually a given graph of y = f(x) with parts asking for sketches of two or three of the listed transformations on the same axes. y = (1)/(f(x)) is the one students get wrong, because zeros become asymptotes and maxima become minima. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

Q: How does FourtyFive help me practise Graphs with modulus functions?

FourtyFive has 4 Graphs with modulus functions questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Q: Is FourtyFive free for Graphs with modulus functions practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Q: Can I handwrite Graphs with modulus functions answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

2 min read Jump to 4 practice questions

Summary

The absolute value function $|x|$ is defined as $x$ for $x \geq 0$ and $-x$ for $x < 0$ , resulting in a V-shaped graph with its vertex at the origin.
Transformations of the form $y = a|x - h| + k$ involve vertical stretching/reflection ( $a$ ), and translations of the vertex to $(h, k)$ .
To graph $y = |f(x)|$ , keep the parts of $y = f(x)$ above the x-axis and reflect the parts below the x-axis over the x-axis.
To graph $y = f(|x|)$ , keep the part of $y = f(x)$ for $x \geq 0$ and reflect it across the y-axis to create an even function.
Modulus equations and inequalities can be solved graphically by finding intersections or regions between the graphs of the left and right sides.

How it is examined

Usually a given graph of $y = f(x)$ with parts asking for sketches of two or three of the listed transformations on the same axes. $y = \frac{1}{f(x)}$ is the one students get wrong, because zeros become asymptotes and maxima become minima. Modulus inequalities are solved by cases or graphically. 6 to 9 marks across parts.

Key ideas

The graphs of the functions $y = |f(x)|$ and $y = f(|x|)$ , $y = \dfrac{1}{f(x)}$ , $y = f(ax+b)$ , $y = [f(x)]^2$ .
Solution of modulus equations and inequalities.

Linking questions

The five graph types listed are the full set. A transformation not on that list is out of syllabus.

Practice questions

4 questions · 2 medium · 2 hard

Showing 4 of 4

Question 1

MediumPaper 2 · calculator7 marks

(a)

A mathematical model describes the behavior of a system. A function $f$ is defined by $f(x) = \frac{2x-5}{x^2-4}$ , where $x \in \mathbb{R}, x \ne \pm 2$ .

(a) Determine the range of $f$ .

[4]

(b)

A function $h$ is defined by $h(x) = f(|x|) \cdot \sin \beta$ , where $x \in \mathbb{R}, x \ne \pm 2$ and $\beta$ is a constant such that $-\pi < \beta < 0$ .

(b) Find the set of values of $x$ such that $h(x) \le 0$ .

[3]

Question 2

HardPaper 1 · no calculator15 marks

(a)

Consider the function $f(x) = \frac{4x-2}{x^2-x-2}$ , for $x \in \mathbb{R}, x \neq 2, x \neq -1$ .

(a) Express $x^2-x-2$ in the form $(x+h)^2+k$ .

[2]

(b)

(b) Express $f(x)$ in partial fractions.

[3]

(c)

[5]

(d)

(d) Find the area of the region enclosed by the graph of $y = f(|x|)$ , the x-axis and the lines with equations $x = -4$ and $x = -3$ .

[5]

Question 3

MediumPaper 1 · no calculator4 marks

Find the exact value of $\int_0^3 |x-1|\,dx$ .

Question 4

HardPaper 1 · no calculator8 marks

(a)(i)

A function $g$ is defined by $g(x)=\frac{3x+6}{x-2}$ , where $x \in \mathbb{R}, x \neq 2$ .

The graph of $y=g(x)$ has a vertical asymptote and a horizontal asymptote.

(a) (i) Write down the equation of the vertical asymptote.

[1]

(a)(ii)

(ii) Write down the equation of the horizontal asymptote.

[1]

(b)

(b) On a set of axes, sketch the graph of $y=g(x)$ . On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.

[3]

(c)

[1]

(d)

(d) Solve the inequality $\frac{3|x|+6}{|x|-2} > 3$ .

[2]

Every Graphs with modulus functions question, marked for you

Every answer is marked mark by mark, IB-style, and the AI tutor helps when you are stuck.

Where marks are lost

Rounding an intermediate value and then using it.
Answering to the wrong accuracy. Two significant figures, or six, where the rule says exactly or three.
Writing the answer and nothing else.

FAQ

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What does Graphs with modulus functions cover in IB Maths AA?

The absolute value function |x| is defined as x for x ≥ 0 and -x for x < 0, resulting in a V-shaped graph with its vertex at the origin. Transformations of the form y = a|x - h| + k involve vertical stretching/reflection (a), and translations of the vertex to (h, k). To graph y = |f(x)|, keep the parts of y = f(x) above the x-axis and reflect the parts below the x-axis over the x-axis.

Is Graphs with modulus functions SL or HL?

Graphs with modulus functions is HL only. SL students are not examined on it.

How do I revise Graphs with modulus functions for IB Maths AA?

Start from the core idea: the absolute value function |x| is defined as x for x ≥ 0 and -x for x < 0, resulting in a V-shaped graph with its vertex at the origin. In the exam: usually a given graph of y = f(x) with parts asking for sketches of two or three of the listed transformations on the same axes. y = (1)/(f(x)) is the one students get wrong, because zeros become asymptotes and maxima become minima. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

How does FourtyFive help me practise Graphs with modulus functions?

FourtyFive has 4 Graphs with modulus functions questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Is FourtyFive free for Graphs with modulus functions practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Can I handwrite Graphs with modulus functions answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

Graphs with modulus functions: notes and practice questions

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Every Graphs with modulus functions question, marked for you

Where marks are lost

Practise this topic as a session

Questions,
answered.

Start with the IB question
bank built for you.

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Every Graphs with modulus functions question, marked for you

Where marks are lost

Questions,answered.

Start with the IB questionbank built for you.

Questions,
answered.

Start with the IB question
bank built for you.