IB Maths AI HL · Unit 2: Functions
IB Maths AI HL Scaling and Linearising Data Questions
Exam-style IB Maths AI HL scaling and linearising data questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 19 questions
- Paper 1: 19
- 5 easy
- 5 medium
- 5 hard
- 4 starter
- 3 worked examples
Practise Scaling and Linearising Data questions →
AI HL formula booklet
What's examined in AI HL scaling and linearising data
The question bank covers these scaling and linearising data question types (number of questions in brackets):
- Deriving Original Equations (9)
- Transforming Non-linear Models (6)
- Data Interpretation and Fitting (4)
Scaling and Linearising Data worked examples
Worked example 1: Linearising a Power Model · easy
A non-linear relationship between two variables is given by the power model $y = ax^n$. By taking the base-$10$ logarithm of both sides, show algebraically how this relationship can be expressed as a linear equation of the form $Y = mX + c$.
1. Take the logarithm of both sides of the equation: $\log_{10} y = \log_{10}(ax^n)$.
2. Apply the product rule for logarithms ($\log(ab) = \log a + \log b$): $\log_{10} y = \log_{10} a + \log_{10} x^n$.
3. Apply the power rule for logarithms ($\log(x^n) = n\log x$): $\log_{10} y = n\log_{10} x + \log_{10} a$.
4. Map this to the linear equation $Y = mX + c$ to finish the proof. Here, $Y = \log_{10} y$ and $X = \log_{10} x$. The gradient is $m = n$ and the $y$-intercept is $c = \log_{10} a$.
Examiner tip: You must explicitly write out the logarithm laws being used to secure the "show that" method marks. Skipping straight from $\log_{10}(ax^n)$ to $n\log_{10} x + \log_{10} a$ may cost you a reasoning mark.
Worked example 2: Extracting Parameters from a Linearised Model · medium
Data for variables $x$ and $y$ is modelled by the exponential function $y = ab^x$. When $\ln y$ is plotted against $x$, a straight line of best fit is formed with a gradient of $-0.45$ and a $y$-intercept of $5.1$. Find the values of $a$ and $b$.
1. Linearise the exponential model by taking the natural logarithm of both sides: $\ln y = \ln a + x\ln b$.
2. Identify the $y$-intercept from the linearised form: $\ln a = 5.1$.
3. Solve for $a$ by raising $e$ to both sides: $a = e^{5.1}$. This evaluates to $164.021\dots \approx$ $164$.
4. Identify the gradient from the linearised form: $\ln b = -0.45$.
5. Solve for $b$ by raising $e$ to both sides: $b = e^{-0.45}$. This evaluates to $0.63762\dots \approx$ $0.638$.
Examiner tip: Make sure you read the question carefully to see whether natural logarithms ($\ln$) or base-10 logarithms ($\log_{10}$) were used on the axes, as this completely changes the base you use to solve for $a$ and $b$.
Worked example 3: Making Predictions from a Linearised Model · hard
Variables $P$ and $Q$ follow a power model $P = kQ^m$. The linear regression line of $\log_{10} P$ against $\log_{10} Q$ is given by $\log_{10} P = 2.5 \log_{10} Q - 1.2$. Estimate the value of $P$ when $Q = 4$.
1. Substitute $Q = 4$ directly into the regression equation: $\log_{10} P = 2.5 \log_{10}(4) - 1.2$.
2. Calculate the value on the right-hand side using your GDC: $2.5(0.60205\dots) - 1.2 = 1.50514\dots - 1.2$.
3. Simplify the result: $\log_{10} P = 0.30514\dots$
4. Convert from logarithmic form back to exponential form to isolate $P$: $P = 10^{0.30514\dots}$.
5. Evaluate the final result to 3 significant figures: $P \approx 2.019\dots \implies$ $2.02$.
Examiner tip: While you could find $k$ and $m$ first and then substitute $Q=4$ into $P = kQ^m$, it is much faster and less prone to intermediate rounding errors to simply substitute $\log_{10}(4)$ into the linear equation and untangle the logarithm at the very end.
Try these IB Maths AI HL scaling and linearising data questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 4 marks · Paper 1
Use the laws of logarithms to expand \(\ln\left(\frac{A}{B^3}\right)\) into a linear expression.
Question 2 · medium · 5 marks · Paper 1
Data for a cooling cup of coffee is plotted on a semi-log graph of \(\ln(T)\) vs time \(t\).
The line passes through \((0, 4.5)\) and \((10, 3.8)\). Find the gradient and \(Y\)-intercept, and state the exponential cooling model.
Question 3 · hard · 7 marks · Paper 1
A biologist records bacteria growth. At \(t=1\), \(N=40\). At \(t=4\), \(N=1080\). Assume an exponential model \(N = A e^{kt}\). Solve algebraically for exact values of \(A\) and \(k\).
All 19 scaling and linearising data questions with mark schemes →
FAQ
How many IB Maths AI HL scaling and linearising data questions are there?
There are 19 exam-style scaling and linearising data questions in the AI HL question bank (Paper 1: 19), graded 5 easy, 5 medium, 5 hard, 4 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is scaling and linearising data on Paper 1 or Paper 2?
In the question bank these questions are set as Paper 1 questions.
Where can I get the mark schemes?
Open the AI HL 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.
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