A designer reflects the curve $y = \ln x$ in the $x$-axis, and then moves the result $3$ units to the right and $2$ units up. The final curve is $y = g(x)$. Find an expression for $g(x)$. [2] Write down the equation of the vertical asymptote of the curve $y = g(x)$. [1] Find the $x$-coordinate of the point where the curve crosses the $x$-axis. [2]

after the reflection: $y = -\ln x$ (A1) $g(x) = 2 - \ln(x - 3)$ A1 $x = 3$ A1 $\ln(x - 3) = 2$ (M1) $x = 3 + e^{2} = 10.4$ A1

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IB Maths AI SL · Unit 2: Functions

IB Maths AI SL Graphing Linear, Quadratic, Exponential and Logarithmic Functions Questions: graphing linear, quadratic, exponential and logarithmic functions

Updated October 2026

Exam-style IB Maths AI SL graphing linear, quadratic, exponential and logarithmic functions questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

85 questions
Paper 1: 49
Paper 2: 36
24 easy
31 medium
20 hard
10 very hard
3 worked examples

Practise Graphing Linear, Quadratic, Exponential and Logarithmic Functions questions →

🖩 How do I do this on my calculator? Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire.

What you need to know

The three quadratic forms (standard, vertex, factored) each carry an M-mark. SL AI usually asks you to model projectile height, profit maximisation, or a bridge arch. Quadratic functions — vertex, roots, and applications overview →

Slope, intercept, and interpreting the gradient in context. SL AI dresses linear models as taxi fares, phone plans, and rental costs — always ask what the gradient MEANS in the real-world context. Linear functions and modelling overview →

What's examined in AI SL graphing linear, quadratic, exponential and logarithmic functions

The question bank covers these graphing linear, quadratic, exponential and logarithmic functions question types (number of questions in brackets):

Graph properties & sketching (70)
Graph intersections & solutions (12)
Equations from graphs/context (3)

Key formulas

Axis of symmetry of a quadratic: $x = - \frac{b}{2 a}$

In the same notation as the IB formula booklet. All AI SL formulas →

Graphing Linear, Quadratic, Exponential and Logarithmic Functions worked examples

Worked example 1: Intercepts of a linear graph · easy

Consider $y = 2 x - 8$ . Find the exact coordinates of the $x$ - and $y$ -intercepts.

Solution

5 steps · 1 to fill in

Examiner tip: Always give intercepts as ordered

(x, y)

pairs unless the question specifically asks for a single coordinate.

Worked example 2: Vertex and y-intercept of a parabola · medium

The parabola has equation $y = - x^{2} + 6 x - 5$ . Find the exact coordinates of its vertex and $y$ -intercept.

Solution

5 steps · 1 to fill in

Examiner tip: Evaluating

- (3)^{2}

: square FIRST (

9

), then apply the minus (

- 9

). A common error is to write

+ 9

Worked example 3: Horizontal asymptote and x-intercept of an exponential · hard

For $f (x) = 2^{x} - 8$ , find the equation of the horizontal asymptote and the exact coordinates of the $x$ -intercept.

Solution

5 steps · 1 to fill in

Examiner tip: For

y = a^{x} + k

, the horizontal asymptote is always

y = k

and the graph never crosses it.

Play with it Roots and y-intercept: drag the sliders

Change a, p and q so that y = a(x − p)(x − q) crosses the x-axis at −1 and 3 and the y-axis at −6. Then expand your answer into the form y = ax² + bx + c.

Want more? Open the full Grapher.

Try these IB Maths AI SL graphing linear, quadratic, exponential and logarithmic functions questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 4 marks · Paper 1

Let $f (x) = 0.5 x^{3} - 3 x^{2} + 4 x + 2$ , for $- 1 \leq x \leq 5$ .

Use your graphic display calculator to find the coordinates of the local maximum and local minimum points. [2]
Sketch the graph of $y = f (x)$ for $- 1 \leq x \leq 5$ , labelling the turning points and the $y$ -intercept. [2]

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 2

A designer reflects the curve $y = \ln x$ in the $x$ -axis, and then moves the result $3$ units to the right and $2$ units up. The final curve is $y = g (x)$ .

Find an expression for $g (x)$ . [2]
Write down the equation of the vertical asymptote of the curve $y = g (x)$ . [1]
Find the $x$ -coordinate of the point where the curve crosses the $x$ -axis. [2]

Full worked solution Free, no sign-in

after the reflection: $y = - \ln x$ (A1)
$g (x) = 2 - \ln (x - 3)$ A1
$x = 3$ A1
$\ln (x - 3) = 2$ (M1)
$x = 3 + e^{2} = 10.4$ A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 8 marks · Paper 2

The height, $h$ metres, of a ball $t$ seconds after it is thrown is modelled by $h (t) = - 4.9 t^{2} + 14 t + 1.5$ , for $0 \leq t \leq T$ , where $h (T) = 0$ .

Find the value of $T$ . [1]
Find the maximum height of the ball and the time at which it occurs. [2]
Sketch the graph of $h$ against $t$ , labelling the intercepts and the maximum point. [2]
Find the length of time for which the ball is at least $8$ m above the ground. [3]

Attempt it and see the mark scheme →

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All 85 graphing linear, quadratic, exponential and logarithmic functions questions with mark schemes →

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FAQ

How many IB Maths AI SL graphing linear, quadratic, exponential and logarithmic functions questions are there?

There are 85 exam-style graphing linear, quadratic, exponential and logarithmic functions questions in the AI SL question bank (Paper 1: 49 · Paper 2: 36), graded 24 easy, 31 medium, 20 hard, 10 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is graphing linear, quadratic, exponential and logarithmic functions on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 49 · Paper 2: 36. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL 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.

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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IB Maths AI SL Graphing Linear, Quadratic, Exponential and Logarithmic Functions Questions: graphing linear, quadratic, exponential and logarithmic functions

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What you need to know

What's examined in AI SL graphing linear, quadratic, exponential and logarithmic functions

Key formulas

Graphing Linear, Quadratic, Exponential and Logarithmic Functions worked examples

Worked example 1: Intercepts of a linear graph · easy

Solution

Worked example 2: Vertex and y-intercept of a parabola · medium

Solution

Worked example 3: Horizontal asymptote and x-intercept of an exponential · hard

Solution

Try these IB Maths AI SL graphing linear, quadratic, exponential and logarithmic functions questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 5 marks · Paper 2

Full worked solution Free, no sign-in

Question 3 · hard · 8 marks · Paper 2

Revise graphing linear, quadratic, exponential and logarithmic functions for the exam

FAQ

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