Piecewise and Other Models (AI SL)
Not every real-world relationship is a single smooth curve. This page covers direct and inverse variation - quantities that scale together or against each other - and piecewise models, where a different rule applies on different parts of the domain, like a tax band that only kicks in above a threshold. It's part of the broader Other Models topic.
24 questions on this sub-topic.
Key relationships
Covered under IB syllabus reference SL2.6: applying the modelling process (fit a function to data or a description, use it to predict, comment on validity) to the function types you already know, including piecewise-defined models built from them. Neither variation form below appears in the formula booklet - you set them up from the description each time.
Direct variation
\(y = kx\)
\(y\) grows in proportion to \(x\). Find \(k\) by substituting one known pair of values, then use the model for any other input.
Inverse variation
\(y = \dfrac{k}{x}\)
\(y\) shrinks as \(x\) grows. The inverse-square version \(y=\dfrac{k}{x^2}\) works the same way, just with \(x^2\) in the denominator.
Need the full syllabus wording and the underlying function types? See Other Models.
Worked examples
The time \(T\) to complete a job varies inversely with the number of workers \(n.\) 6 workers take 8 hours.
(a) Find the model \(T(n).\)
(b) Find the time for 4 workers.
Worked solution
(a) Model. Inverse variation means \(T=\dfrac{k}{n}\). Use \(n=6,\ T=8\): \(8=\frac{k}{6}\Rightarrow k=48,\) A1
so \(T=\dfrac{48}{n}\). A1
(b) Time for 4 workers. \(T=\dfrac{48}{4}=12\) hours. A1
Income tax: 0% on the first $10,000, then 20% on income above that.
(a) Write the tax \(T(x)\) for income \(x>10\,000.\)
(b) Find the tax on an income of $35,000.
Worked solution
(a) Tax formula. The first \($10\,000\) is untaxed; \(20\%\) applies only to the excess \((x-10000)\): \(T(x)=0.20(x-10000),\quad x>10000.\) A1
(b) Tax on \($35\,000\). \(T(35000)=0.20(35000-10000)=0.20(25000)=$5000.\) A1
The kinetic energy \(E\) varies directly with the square of speed \(v.\) At \(v=10,\ E=250\) J.
(a) Find the model.
(b) Find \(E\) when \(v=16.\)
Worked solution
(a) Model. \(E\) varies with \(v^2\): \(E=kv^2\). Use \(v=10,\ E=250\): \(250=k(10)^2=100k\Rightarrow k=2.5,\) A1
so \(E=2.5v^2\). A1
(b) Value at \(v=16\). \(E=2.5(16)^2=2.5(256)=640\text{ J}.\) A1
Common mistakes
- Substituting into the wrong piece of a piecewise model. Always check the input against the stated interval boundaries first - using the wrong formula gives a plausible-looking but wrong answer.
- Confusing direct and inverse variation. "Varies directly" means \(y=kx\) (both grow together); "varies inversely" means \(y=\dfrac{k}{x}\) (one grows as the other shrinks). Misreading the wording flips the whole model.
- Applying the tax rate to the whole income, not just the excess. In a banded model like \(T(x)=0.20(x-10000)\), the rate only applies above the threshold - forgetting to subtract the threshold first overstates the answer.
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24 piecewise-and-variation questions, marked instantly like the real exam.
Quick answers
What is a piecewise model?
A piecewise model uses a different formula on different parts of the domain, such as a tax rate that changes above a threshold. Always check which interval the input falls into before substituting.
What is the difference between direct and inverse variation?
Direct variation is \(y=kx\), where \(y\) increases as \(x\) increases. Inverse variation is \(y=\dfrac{k}{x}\), where \(y\) decreases as \(x\) increases. In both, \(k\) is a constant found by substituting one known pair of values.