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Functions - IB Questionbank | RevisionDojo

Practice Functions with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 2

A particle moves along a straight line with displacement, $s(t)$ , in meters, from a fixed point at time $t$ seconds, modeled by

s(t)=5+4 e^{-0.03 t} \sin \left(\frac{\pi}{10} t\right)-3 \cos \left(\frac{\pi}{10} t+\frac{\pi}{6}\right)

for $0 \leq t \leq 30$ . The particle is at risk of collision when $|s(t)| \geq 6$ .

Find the initial displacement of the particle.

[1]

Write down the period of the oscillatory components.

[2]

Express the oscillatory part of $s(t)$ as a single cosine function with time-dependent amplitude.

[5]

Find the first time the particle is at risk of collision.

[4]

The velocity of the particle is $v(t)=s^{\prime}(t)$ . Find the maximum acceleration of the particle in the first 30 seconds, correct to three significant figures.

[5]

Question 2

SLPaper 1

The number of bacteria in a Petri dish is modelled by the function $N(t)=75 \times 2^{0.5 t}, t \geq 0$ where $N$ is the number of bacteria and is the time in hours.

Write down the number of bacteria in the Petri dish at $t=0$ .

[2]

Calculate the number of bacteria present after 10 hours.

[2]

Calculate the time, in hours, for number of bacteria to reach 10000.

[2]

Question 3

SLPaper 1

A hot winter soup has just been removed from the stove and is left outside to cool. The soup's temperature can be modelled by the function $T(t)=a+b\left(k^{-t}\right)$ where $t$ is the time, in minutes, since the soup was removed from the stove. The temperature outside is $12^{\circ} \mathrm{C}$ .

Write down the value of a and explain why it has this value.

[1]

Initially the temperature of the soup is $95^{\circ} C$ . Find the value of $b$ .

[2]

After two minutes, the temperature of the soup is $60^{\circ} \mathrm{C}$ . Find the value of $k$ .

[2]

After 15 minutes the soup is put into the fridge.

Calculate the temperature of the soup when it is put into the fridge.

[2]

Question 4

HLPaper 2

The point $A(-1,3)$ lies on the curve $y=f(x)$ .

Find the coordinates of the corresponding point under $y=2 f(x-3)+4$ .

[4]

Question 5

SLPaper 2

The front view of the edge of a water tank is drawn on a set of axes below. The edge is modelled by $y=a x^{2}+c$

Point $P$ has coordinates $(-4,4)$ , point $O$ has coordinates $(0,0)$ and point $Q$ has coordinates $(4,4)$ .

Find the value of $c$ and of $a$ .

[2]

Hence, write down the equation of the quadratic function which models the edge of the water tank.

[1]

Given that 1 unit represents 1 m , find the width of the water tank when its height is 2.25 m .

[3]

Question 6

HLPaper 1

The point $A(1,0.5)$ lies on the curve $y=f(x)$ . Write down the coordinates of the corresponding point under the following transformations.

$y=f(x)+5$

[1]

$y=5 f(x)$

[1]

$y=f(x+5)$

[1]

$y=f(5 x)$

[1]

Question 7

SLPaper 2

The average temperature of a city, $C$ , in degrees Celsius, fluctuates throughout a year and can be modelled by the function $C(t)=a \sin (k t)+b$ where $t$ is the elapsed time, in weeks, since the start of the year. The average temperature of the city in week 4 is 27 degrees Celsius and in week 28 it is 12 degrees Celsius.

Find the value of $k$ , assuming there are 52 weeks in a year.

[1]

Write down two equations connecting $a$ and $b$ and find their values.

[4]

Calculate how many weeks of the year that they have to be careful about the food freezing. Give your answer to the nearest integer.

[2]

Question 8

HLPaper 2

The following data represents the population of a certain species of fish in a lake over a period of 10 years. The population is believed to grow exponentially.

Given the data below, convert the results into a suitabale data set for a semi-log graph with base 10.

Year	Population
1	150
2	220
3	320
4	470
5	680
6	980
7	1400
8	2000
9	2900
10	4200

[2]

Using the semi-logarithmic graph, determine the equation of the exponential growth model for the population.

[2]

Question 9

SLPaper 2

In a biology lab, the population of a certain bacteria is observed to grow exponentially. The population ${P(t)}$ in number of bacteria at time ${t}$ hours is modeled by the equation:

${P(t) = 500e^{0.3t}}$

Find the initial population of the bacteria.

[2]

Determine the population after 10 hours.

[2]

Find the time required for the population to double.

[2]

Question 10

HLPaper 2

A city planner is analyzing the linear function representing the cost of public transportation, given by $h(x) = \sqrt{x + 1}$

This function is transformed to produce a new function $j(x)$ .

Write down the equation of $j(x)$ if it represents $h(x)$ reflected in the x-axis.

[2]

Write down the equation of $g(x)$ if it represents $h(x)$ reflected in the y-axis.

[2]

Describe the transformation applied to $h(x) = -x + 1$ to obtain $f(x) = -\sqrt{-x + 1}$ .

[2]

Functions Questionbank