Mathematical Modeling · 8.1.2
Modelling real-life problems with functions
This standard asks students to turn a real-life situation into a linear, quadratic, or exponential function, using given information to determine the unknown constants. Students then use the function to solve the problem (e.g.
find a maximum, a rate, or a future value) and clearly communicate each stage of the modelling process, including assumptions and verification.
The official learning standard (8.1.2)
“Solve real life problems through mathematical modeling which involves the following functions: (i) Linear (ii) Quadratic (iii) Exponential and communicate the mathematical modeling process implemented.”
What it means
This standard asks students to turn a real-life situation into a linear, quadratic, or exponential function, using given information to determine the unknown constants. Students then use the function to solve the problem (e.g.
find a maximum, a rate, or a future value) and clearly communicate each stage of the modelling process, including assumptions and verification.
How it is examined
Mathematical modelling with functions is examined mainly in Paper 2, as an extended structured or open-ended question set in a real-life context (e.g. profit, population, height of a thrown object).
Students must form the function from given data, solve for the required value, and explain the modelling process; Paper 1 may test recognising which function type suits a given scenario.
Worked example
A ball is thrown vertically upward from ground level. Its height h metres above the ground after t seconds is believed to follow a quadratic model h(t) = at² + bt.
Given h(1) = 15 and h(2) = 20, form the model, verify it using h(3) = 15, then find the maximum height reached by the ball.
- Since the ball starts at ground level, h(0) = 0, so the model is h(t) = at² + bt (no constant term).
- Substitute t = 1: a + b = 15.
- Substitute t = 2: 4a + 2b = 20 → 2a + b = 10.
- Solve simultaneously: (2a + b = 10) − (a + b = 15) gives a = -5. Then b = 15 − (-5) = 20.
- Model: h(t) = -5t² + 20t.
- Verify at t = 3: h(3) = -5(9) + 20(3) = -45 + 60 = 15 (matches the given data), so the model is confirmed.
- Maximum height occurs at t = -b/(2a) = -20/(2×-5) = 2 seconds.
- h(2) = -5(4) + 20(2) = -20 + 40 = 20.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I know which type of function to use for modelling?
Look at how the quantity changes: if it increases or decreases at a constant rate, use linear; if it rises then falls (like the height of a thrown object or an area problem), use quadratic; if it grows or decays by a constant percentage each period (like population or radioactive decay), use exponential.
What does communicate the mathematical modeling process actually require me to write?
You should clearly show each stage: state the real-life problem, list your assumptions, form the function using the given information, solve it, then verify your answer makes sense (e.g. checking against another data point or the context) and interpret the result in the original situation, not just as numbers.
Do I need to memorise new formulas for this topic?
No new formula is introduced. You apply what you already know about linear equations (y = mx + c), quadratic equations (y = ax² + bx + c, including the vertex), and exponential functions (y = abx).
The skill being tested is choosing the right function, fitting it to given information, and explaining your reasoning.