Linear Models (AI HL)
A linear model \(f(x) = mx + c\) is the simplest way to describe a quantity that changes at a constant rate - a fixed starting value \(c\) plus a steady rate of change \(m\) per unit of \(x\). This page covers writing a linear model from a real context and handling piecewise versions, where the rate changes at a threshold. It's part of the broader Linear, Quadratic & Cubic Models topic.
11 questions on this sub-topic.
The model and setting it up
Covered under IB syllabus reference SL2.5, which sets out linear, quadratic and cubic modelling as one family of function types. The linear form itself is in the formula booklet; turning a context into that form is the skill this sub-topic tests.
Linear model
\(f(x)=mx+c\)
In the formula booklet. \(m\) is the constant rate of change, \(c\) is the fixed value when \(x=0\).
Setting up a linear model
Identify the fixed starting value (\(c\)) and the constant rate of change (\(m\)) from the context, then write \(f(x)=mx+c\). For a piecewise version, repeat this for each interval where the rate changes.
Need the full syllabus wording, the formula reference table, and GDC regression steps? See Linear, Quadratic & Cubic Models.
Worked examples
A laptop bought for \$1200 loses \$200 of value each year (straight-line).
(a) Write the value \(V(t)\) after \(t\) years.
(b) Find when it reaches zero value.
Worked solution
(a) \(V(t) = 1200 - 200t.\) M1 A1
(b) \(1200 - 200t = 0 \Rightarrow t\) M1
\(= 6\) years. A1
A phone plan charges \$0.15 per minute for the first 100 minutes, then \$0.10 per minute thereafter.
(a) Write the cost \(C(m)\) as a piecewise function.
(b) Find the cost of 250 minutes.
Worked solution
(a) \(C(m) = \begin{cases} 0.15m, & 0 \le m \le 100 \\ 15 + 0.10(m-100), & m > 100 \end{cases}\) M1 A1
(b) \(C(250) = 15 + 0.10(150)\) M1
\(= $30.\) A1
Common mistakes
- Swapping the rate and the fixed value. \(m\) is what changes each step, \(c\) is the one-off starting amount - mixing them up (e.g. writing \(200 - 1200t\) instead of \(1200-200t\)) gives a model with completely the wrong shape.
- Using the wrong piece of a piecewise model. Before substituting, check which interval the input actually falls in - plugging 250 minutes into the "first 100 minutes" rule instead of the one for minutes above 100 gives an answer that's simply wrong, not just imprecise.
- Trusting the model outside its sensible range. A straight-line depreciation model eventually gives a negative value, which makes no sense for an asset's worth - the model is only valid up to the point it hits zero (or whatever the context allows).
Ready to practise properly?
10 linear-model questions, marked instantly like the real exam.
Quick answers
How do I set up a linear model from a word problem?
Identify the fixed starting value (\(c\)) - the value when the input is zero - and the constant rate of change (\(m\)) from the context, then write \(f(x) = mx + c\).
What is a piecewise linear model?
A model built from two or more linear pieces, each valid over its own interval of the input, such as one rate for the first 100 units and a different rate after that. Always check which piece applies before substituting a value in. For calculator steps, see Using your GDC on the full topic page.