HL Modelling (half life, sinusoidal w/ phase shift, logistic, log, piecewise): notes and practice questions
- Mathematical model: simplifies real-world situations for mathematical description and prediction.
- Initial Value: starting value when independent variable (e.g., ) is zero.
- Domain Restrictions: real-world context limits mathematical domain (e.g., ).
- Extrapolation: making unreliable predictions outside the known data range.
- Exponential models (half-life): or .
- Initial value for exponential models: .
- Half-life (base ): .
- Half-life (base ): .
- Sinusoidal models: or .
- Amplitude (): .
- Period (degrees): .
- Period (radians): .
- Phase Shift (): horizontal translation.
- Principal Axis (): .
- Logarithmic model form: (for ).
- Linearisation of Exponential (): (plot vs ).
- Linearisation of Power (): (plot vs ).
- Logistic model form: .
- Limiting Capacity (): horizontal asymptote at .
- Initial Value for logistic model (): .
- Logistic graphs have no roots.
- Piecewise models: constructed from multiple functions, each for a specific domain interval.
- Pay attention to inequality symbols ( vs ) at piecewise boundaries.
- GDC: Check angle settings (Degrees/Radians) for sinusoidal models.
- GDC: Use "Intersect" tool to solve for target values.
How it is examined
Logistic models are a recurring Paper 2 and Paper 3 context because they give a rich set of one-mark interpretation questions: initial value, carrying capacity, when growth is fastest. Continuity of a piecewise model is a clean two or three mark algebra step. The last clause matters most for question generation: a paper may hand HL students a model they have never seen and ask them to use it, so an unfamiliar functional form is not automatically out of syllabus at HL.
The logistic model and the general sinusoidal form.
- Exponential models to calculate half-life.
- Natural logarithmic models, .
- Sinusoidal models, .
- Logistic models, , with .
For piecewise models, parameters may need to be found that make the function continuous. The guide's example: find to make continuous. The formal definition of continuity is not required.
Linking questions
- Other contexts, sinusoidal: waxing and waning of the moon, rainfall patterns, temperature, movement of bridges and buildings, pH scale, Richter scale, sound intensity, brightness of stars.
- Other contexts, piecewise: income taxes and taxi fares, friction, mobile phone plans, depth of a pool as a function of distance from the deep end, postage rates, stock prices, parachuting before and after the chute opens, Hooke's law, shapes of buildings, horizontal distance from a wall to a curved object.
- Links to other subjects: half-life (chemistry and physics); AC circuits and waves (physics); the Gini coefficient and the Lorenz curve, progressive, regressive and proportional taxes, the J-curve (economics).
- TOK: is there a hierarchy of areas of knowledge in terms of usefulness in solving problems?
- Enrichment only, so not examinable: for the population equation with at , the solution is the logistic equation with .
Practice questions
1 question · 1 hardQuestion 1
HardPaper 2 · calculator17 marksEmily and Liam are researching the adoption of smart home devices in a specific region to create a model predicting future usage. They collect the following data:
| Year | Years after 2000 () | Number of devices (in thousands) () |
|---|---|---|
| 2000 | 0 | 10 |
| 2005 | 5 | 150 |
| 2010 | 10 | 400 |
| 2015 | 15 | 750 |
| 2020 | 20 | 1000 |
| 2025 | 25 | 1100 |
Emily proposes the number of devices can be modelled using quadratic regression to find a function of the form , where is the number of years after 2000.
Find the equation of Emily's model.
Emily finds the coefficient of determination for her model is to five significant figures.
State whether the coefficient of determination supports Emily's proposal. Justify your answer.
Comment on the validity of Emily's model with reference to one of the parameters in the equation.
(i) Find the value of and interpret this value in context.
(ii) By considering the changes in device adoption in the table, use the value found in part (d)(i) to comment on the validity of Emily's model.
Liam proposes that the device adoption instead follows a logistic model of the form
where is the number of years after 2000 and is the number of devices in thousands.
State a reason why it may be valid to use Liam's proposal to predict future device adoption.
(i) Find .
(ii) Hence find the year, according to Liam's model, during which the greatest device adoption growth rate occurred.
Use your GDC's regression features (e.g., QuadraticReg) to find the coefficients , , and . Ensure you input the 'Years after 2000' as your -values and 'Number of devices (in thousands)' as your -values.
Recall what a coefficient of determination () value close to 1 indicates about the model's fit to the data.
Consider the real-world implications of the values of or in the context of device adoption. Can the number of devices be negative, or can it decrease indefinitely?
For (i), differentiate with respect to to find , then substitute . The derivative represents the rate of change. For (ii), compare the model's predicted rate of change with the actual data in the table, especially for the period around .
Think about the long-term behaviour of logistic models, especially in the context of growth phenomena like technology adoption.
For (i), use the chain rule or quotient rule to differentiate . Remember that . For (ii), the maximum growth rate for a logistic function occurs when , where is the coefficient of in the denominator.
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Where marks are lost
- Answering to the wrong accuracy. Two significant figures, or six, where the rule says exactly or three. Common wherever a GDC's full decimal display gets copied straight down.
- Rounding an intermediate value and then using it in a later part. Costs a mark every time, and AI's multi-part modelling questions give it more chances to happen than AA's shorter, more self-contained ones.
- Writing the answer and nothing else, where the mark scheme has an explicit M1 rather than an implied one. A bare answer cannot score full marks there.
