The height of a baseball after it is hit by a bat is modelled by the quadratic function $h(t) = -4.8t^2 + 21t + 1.2$ , where $h(t)$ is the height in metres above the ground and $t$ is the time in seconds after the ball was hit. Write down the height of the ball above the ground at the exact instant it is hit. Find the maximum height reached by the baseball. Use your GDC to find the value of $t$ when the ball hits the ground.

Substitute $t=0$ : $h(0) = 1.2\text{ m}$ . A1 Vertex: $t = \frac{-21}{2(-4.8)}$ . (M1) $t = 2.1875\text{ s}$ . A1 Substitute: $h(2.1875) = -4.8(2.1875)^2 + 21(2.1875) + 1.2 \approx 24.2\text{ m}$ ( $24.168\dots$ ). A1 Set $h(t) = 0$ : $-4.8t^2 + 21t + 1.2 = 0$ ( CG50: Equation $\to$ Polynomial ). (M1) $t \approx 4.43\text{ s}$ ( $4.431\dots$ ); reject the negative root $t \approx -0.0564$ . A1

IB Maths AI SL · Unit 2: Functions

IB Maths AI SL Applications of Functions Questions

Updated October 2026

Exam-style IB Maths AI SL applications of functions questions with worked solutions. Start with the 5 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.

40 questions
Paper 1: 18
Paper 2: 22
10 easy
13 medium
9 hard
8 very hard
5 worked examples

Practise Applications of Functions questions →

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

What's examined in AI SL applications of functions

The question bank covers these applications of functions question types (number of questions in brackets):

Analyzing Given Models (23)
Constructing Models (12)
Optimisation Problems (5)

Applications of Functions worked examples

Worked example 1: Evaluating a function at a given value · easy

A function is defined as $f (x) = 3 x^{2} - 5 x + 2$ . Calculate the exact value of $f (- 4)$ .

Solution

4 steps · 1 to fill in

Examiner tip: Always use parentheses around negative numbers when substituting, especially before squaring — this stops your calculator from evaluating

- 4^{2}

- 16

Worked example 2: Solving g(x) = 7 graphically · medium

Let $g (x) = \frac{12}{x} + x$ for $x > 0$ . Using your GDC, find the exact value(s) of $x$ for which $g (x) = 7$ .

Solution

5 steps · 1 to fill in

Examiner tip: SL AI students are strongly encouraged to use the GDC graphing/numerical solver for roots and intersections rather than relying on algebra alone.

Worked example 3: Intersecting linear and exponential graphs · hard

Consider $f (x) = 2^{x}$ and $g (x) = 10 - 2 x$ . Find the coordinates of the point of intersection to 3 s.f.

Solution

5 steps

Examiner tip: Equations mixing exponential and polynomial terms cannot be solved with basic algebra. Reach straight for the GDC intersection tool.

Worked example 4: Solving a linear cost model · easy

A taxi company models the cost $C$ (in euros) as $C (d) = 3.50 d + 4.00$ , where $d$ is distance in km. Calculate the exact distance for a total cost of $€ 32.00$ .

Solution

5 steps · 1 to fill in

Examiner tip: In a linear cost model

y = m x + c

: the constant

c

is the fixed initial fee, and the coefficient

m

is the per-unit rate.

Worked example 5: Break-even point for a manufacturer · medium

Daily cost to produce $x$ water bottles is $C (x) = 450 + 3.5 x$ ; daily revenue is $R (x) = 12.5 x$ . Find the minimum number of bottles that must be sold to make a strictly positive profit.

Solution

5 steps

Examiner tip: Watch for "strictly positive": selling exactly 50 gives zero profit (break-even). The answer is the next integer, 51.

Try these IB Maths AI SL applications of 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 · 5 marks · Paper 1

The value of a specific industrial machine, $V$ , in euros, can be modelled as a linear function of time, $t$ , in years since it was purchased. The model is given by $V (t) = 12 500 - 1100 t$ .

Write down the initial purchase price of the machine.
Calculate the value of the machine exactly 4 years after purchase.
Find the exact time, $t$ , when the machine loses all of its value ( $V = 0$ ).

Attempt it and see the mark scheme →

Question 2 · medium · 6 marks · Paper 1

The height of a baseball after it is hit by a bat is modelled by the quadratic function $h (t) = - 4.8 t^{2} + 21 t + 1.2$ , where $h (t)$ is the height in metres above the ground and $t$ is the time in seconds after the ball was hit.

Write down the height of the ball above the ground at the exact instant it is hit.
Find the maximum height reached by the baseball.
Use your GDC to find the value of $t$ when the ball hits the ground.

Full worked solution Free, no sign-in

Substitute $t = 0$ : $h (0) = 1.2 m$ . A1
Vertex: $t = \frac{- 21}{2 (- 4.8)}$ . (M1)
$t = 2.1875 s$ . A1
Substitute: $h (2.1875) = - 4.8 (2.1875)^{2} + 21 (2.1875) + 1.2 \approx 24.2 m$ ( $24.168 \dots$ ). A1
Set $h (t) = 0$ : $- 4.8 t^{2} + 21 t + 1.2 = 0$ (CG50: Equation $\to$ Polynomial). (M1)
$t \approx 4.43 s$ ( $4.431 \dots$ ); reject the negative root $t \approx - 0.0564$ . A1

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

Question 3 · hard · 6 marks · Paper 1

A biologist uses the following quadratic model to estimate the growth rate of a specific algae culture, $y$ , at varying temperatures, $x$ (in degrees Celsius): $y = - 0.45 x^{2} + 3.2 x - 1.7$

Using your GDC’s maximum tool, find the temperature $x$ that produces the maximum growth rate, and state this maximum rate.
Using your GDC, find the two temperatures where the growth rate $y$ is exactly zero.

Attempt it and see the mark scheme →

Free account — sign in to download Topic PDF: 7 questions with worked answers · 6 pages · 265 KB

All 40 applications of functions questions with mark schemes →

Save your progress and get 5 more questions on every topic, checked instantly. Free account, no card. Join free

Revise applications of functions for the exam

FAQ

How many IB Maths AI SL applications of functions questions are there?

There are 40 exam-style applications of functions questions in the AI SL question bank (Paper 1: 18 · Paper 2: 22), graded 10 easy, 13 medium, 9 hard, 8 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is applications of functions on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 18 · Paper 2: 22. 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.

Join free

Already have an account? Sign in

IB Maths AI SL Applications of Functions Questions

Try it first

What's examined in AI SL applications of functions

Applications of Functions worked examples

Worked example 1: Evaluating a function at a given value · easy

Solution

Worked example 2: Solving g(x) = 7 graphically · medium

Solution

Worked example 3: Intersecting linear and exponential graphs · hard

Solution

Worked example 4: Solving a linear cost model · easy

Solution

Worked example 5: Break-even point for a manufacturer · medium

Solution

Try these IB Maths AI SL applications of functions questions

Question 1 · easy · 5 marks · Paper 1

Question 2 · medium · 6 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 1

Revise applications of functions for the exam

FAQ

What else is free here

More AI SL Unit 2 topics