Modelling Change with Linear and Quadratic Functions
The modelling cycle
Mathematical modelling is the craft of turning a messy real situation into a clean equation, using that equation to answer a question, and then asking honestly how well the answer fits reality. It is less about a single calculation and more about a cycle: formulate a model, solve it, interpret the solution in the words of the original problem, and finally evaluate whether the model was good enough and report what you found. This unit applies that cycle to problems involving change, including money, choosing in each case between a linear and a quadratic function.
Choosing linear or quadratic
The first decision is the most important one: which kind of function fits the situation. The test is how the quantity changes. If something grows or shrinks by the same amount every step, the rate of change is constant and a linear function, a straight line, is the right tool. If instead the rate of change itself changes, or the quantity is built from an area or a product of two things that both vary, a squared term appears and a quadratic function fits. Getting this choice right at the start saves all the work that follows.
A linear model for everyday money
Linear models suit most everyday money problems. Suppose a phone plan charges $15 a month plus 20 cents for each gigabyte of data. The cost in dollars is 15 plus 0.2 times the number of gigabytes, written as cost = 0.2x + 15. The 15 is the fixed starting value and the 0.2 is the constant rate per gigabyte. At 25 gigabytes the cost is $20. Because every extra gigabyte adds exactly the same 20 cents, the model is linear, and its graph is a straight line climbing steadily.
Running a linear model forwards and backwards
Solving a linear model usually means running it forwards or backwards. A savings plan starting at $200 and adding $50 a week is modelled by amount = 50x + 200, where x counts the weeks. Forwards, after 8 weeks the balance is $600. Backwards, asking when the balance first reaches $600 means solving 50x + 200 = 600, which gives x = 8 weeks. Interpreting that solution in context, not as a bare number but as eight weeks of saving, is part of the modelling, not an afterthought.
Quadratic models for area
Quadratic models appear whenever an area or a falling-and-rising motion is involved. Imagine fencing a rectangular garden whose width and length add to 20 metres, so the length is 20 − x when the width is x. The area is width times length, that is x(20 − x), which expands to 20x − x², a quadratic. A table of widths shows the area rising to a peak and falling away again, reaching its largest value of 100 square metres when the width is 10 metres, which makes the garden a square. The peak sits at the vertex of the parabola; here the squared term has a negative coefficient, so the parabola opens downward and the vertex is a maximum rather than a minimum.
Quadratic models for motion
Motion under gravity gives a second classic quadratic. A toy rocket whose height in metres is modelled by h = −5t² + 20t, with t in seconds, climbs and then falls. Evaluating the model at a few times gives heights of 15, 20 and 15 metres at 1, 2 and 3 seconds. The greatest height, 20 metres, occurs at the vertex when t = 2 seconds, and the rocket returns to the ground, height zero, at t = 4 seconds. Each of these numbers answers a real question about the flight, which is what interpreting a solution means.
Evaluate and report: a model knows its limits
The cycle is not finished until you evaluate and report. Evaluating asks whether the model is realistic: the savings line assumes you never miss a week, and the rocket model ignores air resistance, so each answer is only as trustworthy as those assumptions. Reporting means stating clearly what you did, what you found, and where the model might break down. A good model is honest about its own limits, and presenting both the answer and its caveats is the hallmark of real mathematical modelling rather than mere arithmetic.