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Linear Models and Modeling Skills
Using functions to fit real world data and make predictions
Using functions to fit real world data and make predictions
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A mathematical model is an equation or graph that represents a real-world situation and can be used to analyze and make predictions about that situation. Mathematical models may be exact or approximate.
Because real-world scenarios usually involve many variables, we often identify the most important ones and making reasonable assumptions about the rest. A good model simplifies the situation as much as possible without significantly reducing the accuracy of its predictions.
In a mathematical model, constants and coefficients are called parameters. The general shape of a model is given by its family (linear, quadratic, exponential, etc.), but the more specific values (like intercepts, asymptotes, or steepness) are controlled by the parameters.
A linear model is represented by a straight-line graph.
Since a linear model can be defined by one point and a gradient or two points, they are the simplest models to construct. The most common form of a linear model is y=ax+b, where a is the slope and b is the y-intercept.
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Extrapolation is when we predict values beyond the domain of the given points. Extrapolating may work for certain situations, but it does not work for many others. Pay attention to the context of a model when extrapolating and consider whether the observed behavior is likely to change in the long-run.
Your understanding of extrapolation can be tested by questions that ask you to interpret plausible inputs and outputs.