Mathematical Modelling with Functions
Learn to choose and apply appropriate mathematical models using exponential, logarithmic, and trigonometric functions for real-world problems in HSC Advanced Mathematics.
Choosing the Right Model
Mathematical modelling uses functions to represent real-world situations. The key skill is recognising which type of function best fits the data or scenario.
Linear: y = mx + b — constant rate of change (e.g., fixed speed, constant cost per unit).
Quadratic: y = ax2 + bx + c — projectile motion, maximising/minimising area.
Exponential: y = A · ekt — growth/decay (population, radioactive decay, compound interest).
Logarithmic: y = a ln(x) + b — diminishing returns (loudness, earthquake intensity).
Trigonometric: y = A sin(Bx + C) + D — periodic phenomena (tides, temperature, sound).
How to choose: Look at the shape of the data (straight line, curve, oscillating), whether it has a maximum/minimum, whether it levels off, and whether it repeats.
Consider the context: growth/decay suggests exponential; cycles suggest trigonometric.
Exponential and Logarithmic Modelling
Exponential functions model situations where the rate of change is proportional to the current value. The general form is:
y = A · ekt
k > 0 for growth, k < 0 for decay. A is the initial value.
To find the parameters, use known data points. Logarithms are the inverse of exponentials and are used to "undo" exponential equations when solving for time or rate.
Example: A bacteria population doubles every 3 hours. If there are initially 500 bacteria, model the population.
P(t) = 500 · 2t/3, or equivalently P(t) = 500 · e(ln2/3)t ≈ 500e0.231t.
When does P = 8000? Solve: 8000 = 500 · 2t/3, so 16 = 2t/3, t/3 = 4, t = 12 hours.
Trigonometric Modelling
Trigonometric functions model periodic phenomena — situations that repeat at regular intervals. The general sinusoidal model is:
y = A sin(B(x − C)) + D
A = amplitude, B = 2π/period, C = horizontal shift, D = vertical shift (midline).
To build a trigonometric model from data: find the maximum and minimum values to determine A and D, then find the period to determine B, and use a known point to find C.
Example: A tide varies between 1.2m and 3.8m with a period of 12 hours. High tide is at 3am.
A = (3.8 − 1.2)/2 = 1.3, D = (3.8 + 1.2)/2 = 2.5, B = 2π/12 = π/6.
Model: h(t) = 1.3 sin(π/6 (t − 3)) + 2.5, where t is hours after midnight.
Key Vocabulary
Mathematical Model
A function or equation that represents a real-world situation, allowing predictions and analysis.
Exponential Growth/Decay
When a quantity changes at a rate proportional to its current value: y = Aekt.
Period
The length of one complete cycle of a periodic function, given by T = 2π/B.
Amplitude
The maximum displacement from the midline of a sinusoidal function: A = (max − min)/2.
Worked Examples
A radioactive substance has a half-life of 10 years. If there are initially 200g, model the remaining mass and find when 50g remains.
Step 1: M(t) = 200 · (1/2)t/10 = 200 · e−(ln2/10)t.
Step 2: Solve 50 = 200 · (1/2)t/10. So 1/4 = (1/2)t/10, giving t/10 = 2.
Answer: t = 20 years. The substance takes 20 years to decay to 50g (two half-lives).
Temperature in a city varies between 8°C (min in July) and 28°C (max in January). Model the temperature as a function of months after January.
Step 1: A = (28 − 8)/2 = 10, D = (28 + 8)/2 = 18, Period = 12 months, B = 2π/12 = π/6.
Step 2: Max at t = 0 (January), so use cosine: T(t) = 10 cos(πt/6) + 18.
Answer: T(t) = 10 cos(πt/6) + 18, where t is months after January.
The loudness L (in decibels) of a sound is modelled by L = 10 log10(I/I0). If a sound has intensity 1000 times I0, find L.
Step 1: L = 10 log10(1000/I0 × 1/I0) ... no, L = 10 log10(I/I0) = 10 log10(1000).
Step 2: log10(1000) = 3.
Answer: L = 10 × 3 = 30 decibels.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
Which function type best models the height of a ball thrown into the air?
Question 2
A population of 1000 grows at 5% per year. What is the population after 10 years?
Question 3
A sinusoidal function has a maximum of 20 and minimum of 4. What is the amplitude?
Question 4
A substance decays from 80g to 20g in 6 hours. What is the half-life?
Question 5
Which scenario would best be modelled by a trigonometric function?
Key Concepts Summary
- ● Choose models based on the data pattern: linear (constant change), quadratic (parabolic), exponential (growth/decay), trigonometric (periodic).
- ● Exponential models y = Aekt describe growth (k > 0) and decay (k < 0).
- ● Logarithmic models are the inverse of exponential functions and describe diminishing returns.
- ● Trigonometric models y = A sin(B(x − C)) + D capture periodic behaviour with amplitude, period, and phase shift.
- ● Always consider the real-world context and check model predictions against known data.