lesson
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If you raise the price of a movie ticket, you make more money per seat, but fewer people show up. Does your total revenue go up forever in a straight line, or does it hit a peak and drop?
To solve problems like this, we need to choose between two powerful tools: linear models and quadratic models.
This decision is at the heart of the Mathematical Modelling Process: understand the problem โ formulate a model โ solve mathematically โ interpret solutions in context โ evaluate model validity and limitations.
How do we know whether a real-world pattern follows a straight line or curves toward a peak?
Linear vs Quadratic: The Core Difference
Your primary guide is the Rate of Change Clue: a linear model y=mx+c represents a constant rate of change, such as a fixed cost per unit or constant velocity.
In contrast, a quadratic model y=ax2+bx+c represents a uniformly changing rate of change, such as constant acceleration under gravity or declining ticket demand as price increases.
Look for Behaviour Clues and Critical Points: linear functions are strictly monotonic (always increasing or decreasing), whereas any scenario involving a peak, turnaround, or symmetry requires a quadratic model.
๐Interactive diagram
What if you are not given a graph or an equation, but just a raw table of data?
The Differences Test
When your inputs (x) increase by equal steps, you can subtract consecutive outputs (y) to find the first differences.
This is the Finite Differences Clue: when tabular data has equally spaced inputs (ฮx=k), a linear function has constant first differences (ฮy), whereas a quadratic function has constant second differences (ฮ2y=2a).