Functions (AI SL) — IB Math AI SL AI SL Study Guide

For: IB Math AI SL candidates sitting IB Math: Applications & Interpretation SL.

Covers: Linear, quadratic, exponential, logarithmic, and sinusoidal function models, real-world phenomenon modelling, and domain/range derivation from contextual problems for IB Math AI SL exams.

You should already know: IGCSE / pre-DP math.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the IB Math AI SL style for educational use. They are not reproductions of past IBO papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official IBO mark schemes for grading conventions.

1. What Is Functions (AI SL)?

Functions are mathematical relationships that map every input value to exactly one output value, and in IB Math AI SL, you will use them as predictive tools to model real-world scenarios rather than just abstract algebraic objects. The standard notation is $f(x)$, where $x$ is the independent input variable and $f(x)$ is the dependent output variable; common synonyms include "mapping" and "input-output rule". This topic makes up 15-20% of your total exam mark, appearing on both Paper 1 (no calculator) and Paper 2 (calculator allowed), with 6-8 mark extended response questions almost always tied to real-world context.

2. Linear, quadratic, exponential, log models

These four function types are the most commonly tested core models in IB Math AI SL, each designed to fit specific patterns of change:

• Linear models: Follow the form $f(x)=mx+c$, where $m$ is the constant rate of change (slope) and $c$ is the y-intercept (output value when $x=0$). They are used for scenarios with fixed increase or decrease, such as hourly wages, delivery costs with a fixed fee, or distance travelled at constant speed. Worked snippet: A coffee shop charges a $4fixeddeliveryfeeplus$3.50 per iced latte. The total cost for $x$ lattes is $C(x)=3.5x+4$. For an order of 6 lattes, $C(6)=3.5(6)+4=25$, so the total cost is $25.
• Quadratic models: Follow the form $f(x)=a{x}^2+bx+c$ where $a\ne 0$, with a parabolic graph that opens upward if $a>0$ (has a minimum point) or downward if $a<0$ (has a maximum point). They are used for scenarios with a single peak or trough, such as projectile motion, profit vs selling price, or area maximization problems. Worked snippet: The height of a thrown ball in meters is $h(t)=-4.9t^2+12t+1.5$, where $t$ is time in seconds. The maximum height occurs at the vertex: $t=-\frac{b}{2a}=-\frac{12}{2(-4.9)}\approx 1.22s$, and the maximum height is $h(1.22)\approx 8.84m$.
• Exponential models: Follow the form $f(x)=k{a}^x$, where $k$ is the initial output value when $x=0$, $a>1$ indicates exponential growth (e.g. compound interest, bacterial population growth), and $0<a<1$ indicates exponential decay (e.g. radioactive decay, drug concentration in blood). Worked snippet: $1500investedat4.2$A(t) = 1500(1.042)^t$,where$t$istimeinyears.After8years,theinvestmentvalueis$A(8) \approx 1500(1.390) \approx 2085$,sototalinterestearnedis$585.
• Logarithmic models: Follow the form $f(x)=k{\log }_a(x)+c$, and are the inverse of exponential functions. They are used for scenarios where output grows very slowly as input increases, such as the Richter earthquake scale, pH scale, or decibel sound level scale. Worked snippet: Richter magnitude is calculated as $M={\log }_{10}\left(\frac{I}{I_0}\right)$, where $I$ is earthquake intensity and $I_0$ is a reference intensity. An earthquake with $M=7$ is 100 times more intense than an earthquake with $M=5$, since ${\log }_{10}(I_1/I_0)-{\log }_{10}(I_2/I_0)=2⟹I_1/I_2=10^2=100$.

Exam tip: To identify the correct model for a given data set, check: linear models have constant first differences between consecutive outputs, quadratic models have constant second differences, exponential models have constant ratios between consecutive outputs for evenly spaced $x$-values.

3. Sinusoidal models

Sinusoidal models are periodic functions that repeat over a fixed interval (called the period), used for cyclical real-world phenomena such as tide heights, annual temperature fluctuations, the height of a point on a Ferris wheel, or alternating current flow. The general forms you will use for IB AI SL are: $$f(x)=a\sin \left(b(x-h)\right)+k\text{or}f(x)=a\cos \left(b(x-h)\right)+k$$ Where:

• Amplitude $|a|$: Half the distance between the maximum and minimum output values, calculated as $a=\frac{\text{max}-\text{min}}{2}$
• Period $T$: Length of one full cycle, related to parameter $b$ by $T=\frac{2\pi }{|b|}$ (so $b=\frac{2\pi }{T}$)
• Vertical shift $k$: The midline of the function, calculated as $k=\frac{\text{max}+\text{min}}{2}$
• Horizontal shift $h$: Phase shift, moves the entire graph left or right to align with the starting point of the cycle.

Worked example: The height of water in a coastal harbour varies between 0.8m at low tide and 8.2m at high tide, with 12.4 hours between consecutive high tides. At $t=0$, the water is at high tide. Write a sinusoidal model using cosine. Step 1: Calculate amplitude and midline: $a=\frac{8.2-0.8}{2}=3.7$, $k=\frac{8.2+0.8}{2}=4.5$ Step 2: Calculate $b$: $T=12.4$, so $b=\frac{2\pi }{12.4}\approx 0.507$ Step 3: Cosine starts at its maximum value when $h=0$, so no phase shift is needed. Final model: $h(t)=3.7\cos (0.507t)+4.5$. Verify at $t=0$: $h(0)=3.7(1)+4.5=8.2$, which matches the high tide value.

Exam tip: Always choose the function (sine or cosine) that eliminates the need for a phase shift if possible, to avoid calculation errors from the $h$ parameter.

4. Modelling real-world phenomena

Applied modelling is the core focus of the IB Math AI SL course, so most high-mark function questions will require you to build a model from scratch for a real-world context, rather than solving abstract algebraic problems. The standard 5-step process you are expected to follow for 6-8 mark modelling questions is:

1. Define variables explicitly: State what your independent and dependent variables represent, including units (e.g. "let $t$ = time in hours after 12pm, let $T(t)$ = air temperature in °C")
2. Select the appropriate model type: Choose linear, quadratic, exponential, logarithmic, or sinusoidal based on the context description or given data patterns
3. Solve for unknown parameters: Use given data points to calculate the values of constants in your model
4. Make predictions: Use your completed model to answer the question's required prediction (e.g. "find the temperature at 8pm")
5. Comment on reasonableness: State the limitations of your model (e.g. "this exponential growth model is only valid for 12 years, after which food resources will limit population growth")

Worked example: A bacterial colony doubles in size every 2.5 hours, starting with 300 cells at $t=0$. Find the number of cells after 7 hours, and comment on the model's reasonableness. Step 1: Define variables: $t$ = time in hours, $P(t)$ = number of bacterial cells Step 2: Exponential growth model: $P(t)=P_0{a}^t$ Step 3: $P_0=300$, at $t=2.5$, $P=600$, so $600=300a^{2.5}⟹a^{2.5}=2⟹a=2^{1/2.5}\approx 1.3195$. Completed model: $P(t)=300(1.3195{)}^t$ Step 4: At $t=7$: $P(7)=300(1.3195{)}^7\approx 300(6.964)\approx 2089$ cells Step 5: Reasonableness: This model assumes unlimited nutrients and space, so it is only valid for the first 20 hours, after which waste buildup and resource limits will stop exponential growth.

Exam note: You will lose 1 mark per question if you do not explicitly define your variables, even if your model is correct.

5. Domain and range from context

The domain of a function is the set of all valid input values, while the range is the set of all valid output values corresponding to that domain. In abstract math, these are derived from algebraic constraints (e.g. no negative numbers under square roots), but in IB Math AI SL, you will almost always derive them from real-world context rather than pure algebra. Key rules to remember:

• Inputs cannot be negative if they represent time, distance, or counts of physical items (you cannot have -3 cups of coffee or -2 hours)
• Inputs cannot exceed the value where the model stops being valid (e.g. if a ball hits the ground at $t=3.2s$, the domain of the height function is $0\le t\le 3.2$, not all real numbers)
• The range only includes output values that correspond to the valid domain, not the full theoretical range of the function
• Countable items (leaflets, mugs, people) require integer inputs, while continuous measurements (time, temperature) allow real number inputs.

Worked example: The cost of printing $x$ custom posters is $C(x)=0.75x+25$, where the printer requires a minimum order of 20 posters and a maximum order of 300 posters per customer. Find the domain and range of $C(x)$.

• Domain: Minimum order is 20, maximum is 300, and posters are countable so $x$ must be integer: $20\le x\le 300$, $x\in \mathbb{Z}$
• Range: Calculate cost at domain endpoints: $C(20)=0.75(20)+25=40$, $C(300)=0.75(300)+25=250$, so range is $40\le C(x)\le 250$.

Exam trap: 60% of students lose 2 marks per domain/range question by writing the full algebraic domain (e.g. $x\ge 0$) instead of using the context-specific limits given in the problem. Always re-read the problem for explicit constraints before writing your answer.

6. Common Pitfalls (and how to avoid them)

• Wrong move: Using the full algebraic domain/range instead of context-derived values. Why: Students memorize abstract function rules and ignore scenario constraints. Correct move: Always scan for explicit limits (minimum order, time when object hits ground) and implicit limits (no negative time, integer counts) before answering domain/range questions.
• Wrong move: Mixing up the period and $b$ parameter in sinusoidal models, writing $b=\frac{T}{2\pi }$ instead of $b=\frac{2\pi }{T}$. Why: Students rush through formula recall under exam pressure. Correct move: Double check by plugging your $b$ value back in: if $T=12$, $b=\frac{2\pi }{12}=\frac{\pi }{6}$, so when $x=12$, the argument of $\sin /\cos$ is $2\pi$, which is one full cycle, confirming the value is correct.
• Wrong move: Forgetting to define variables when building a context model. Why: Students assume examiners know what $x$ and $f(x)$ represent. Correct move: Write a 1-line definition for both variables immediately when starting a modelling question, including units.
• Wrong move: Rounding parameter values too early when building models, leading to large errors in final predictions. Why: Students round to 2 decimal places early for convenience. Correct move: Keep 4+ significant figures for all parameters during calculation, only round final answers to the required 3 significant figures for IB AI SL.
• Wrong move: Using a linear model for data with a changing rate of change. Why: Students default to the simplest model without checking data patterns. Correct move: Calculate first differences (constant = linear), second differences (constant = quadratic), or output ratios (constant = exponential) to confirm the correct model.

7. Practice Questions (IB Math AI SL Style)

Question 1

The average monthly high temperature in a Mediterranean town varies sinusoidally over the year. The highest average high is 32°C in August (month 8) and the lowest is 10°C in February (month 2). a) Write a sinusoidal model for the average monthly high temperature $T(m)$, where $m$ is the month number (1 = January). [4 marks] b) Use your model to estimate the average high temperature in May (month 5). [2 marks] c) State the domain and range of $T(m)$ for a single calendar year. [2 marks]

Worked Solution 1

a) Step 1: Calculate parameters: max $T=32$, min $T=10$, so amplitude $a=\frac{32-10}{2}=11$, midline $k=\frac{32+10}{2}=21$. Period $T=12$ months, so $b=\frac{2\pi }{12}=\frac{\pi }{6}$. Use cosine with a phase shift of 8 to align the maximum with month 8: $$T(m)=11\cos \left(\frac{\pi }{6}(m-8)\right)+21$$ Verify at $m=8$: $11\cos (0)+21=32$, correct. At $m=2$: $11\cos \left(\frac{\pi }{6}(-6)\right)+21=11\cos (-\pi )+21=-11+21=10$, correct. b) May is $m=5$: $$T(5)=11\cos \left(\frac{\pi }{6}(5-8)\right)+21=11\cos \left(-\frac{\pi }{2}\right)+21=0+21=21°C$$ c) Domain: $1\le m\le 12$, $m$ is integer. Range: $10\le T(m)\le 32$.

Question 2

A street food vendor sells loaded fries, and finds their daily profit $P(x)$ in USD follows the quadratic model $P(x)=-15x^2+330x-1080$, where $x$ is the price per serving in USD. a) Find the price the vendor should charge to maximize daily profit, and the maximum daily profit. [3 marks] b) Find the minimum and maximum price the vendor can charge to earn a positive profit. [3 marks]

Worked Solution 2

a) The quadratic opens downward ($a=-15<0$), so maximum profit occurs at the vertex: $$x=-\frac{b}{2a}=-\frac{330}{2(-15)}=11$$ The vendor should charge $11 per serving. Substitute back to find maximum profit: $$P(11)=-15(11{)}^2+330(11)-1080=-1815+3630-1080=735$$ Maximum daily profit is $735. b) Positive profit requires $P(x)>0$: $$-15x^2+330x-1080>0$$ Divide both sides by -15, reversing the inequality: $$x^2-22x+72<0$$ Factor: $(x-4)(x-18)<0$, so solutions are $4<x<18$. The minimum price is $4,maximumpriceis$18 to earn positive profit.

Question 3

A radioactive isotope has a half-life of 22 years. The initial mass of a sample is 120 grams. a) Write an exponential model for the mass $M(t)$ of the isotope remaining after $t$ years. [3 marks] b) Find the mass remaining after 60 years, correct to 3 significant figures. [2 marks]

Worked Solution 3

a) Exponential decay model for half-life: $M(t)=M_0(0.5{)}^{t/T_{1/2}}$, where $M_0=120g$, $T_{1/2}=22$ years. $$M(t)=120(0.5{)}^{t/22}$$ b) Substitute $t=60$: $$M(60)=120(0.5{)}^{60/22}=120(0.5{)}^{\mathrm{2.727...}}\approx 120(0.1529)\approx 18.3g$$ (3 sf)

8. Quick Reference Cheatsheet

Additional Rules:

1. Domain from context: No negative time/counts, respect explicit minimum/maximum limits, integer inputs for countable items
2. Range from context: Only includes output values corresponding to the valid context domain
3. Modelling workflow: Define variables → select model → solve for parameters → make predictions → state limitations
4. Half-life formula: $M(t)=M_0(0.5{)}^{t/T_{1/2}}$

9. What's Next

This functions topic is the foundation for almost all other applied content in the IB Math AI SL syllabus. You will reuse these model types when studying differential calculus (calculating rates of change for real-world quantities), statistical regression (fitting function models to empirical data sets), and financial mathematics (compound interest, loan repayment, and investment growth models). Understanding how to interpret function parameters in context is also a critical skill for your internal assessment (IA), where you will design and investigate a real-world mathematical problem of your choice.

If you are stuck on any subtopic, from solving for sinusoidal phase shifts to deriving domain and range from complex context, you can ask Ollie, our AI tutor, for personalized explanations, additional practice questions, or step-by-step walkthroughs of past exam problems. You can also find more topic-specific study guides and full timed mock exams on the homepage to build your confidence ahead of your IB Math AI SL exams.