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Linear Models and Modeling Skills
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Mathematical modelling with linear models y=ax+b, using assumptions and parameters to represent real situations, and interpreting or extrapolating predictions beyond the given data when appropriate.
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Mathematical modelling and assumptions
SL AI 2.5
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.
Linear models
SL AI 2.5
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.
Extrapolation
SL AI 2.5
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.
Example
Between the ages of 5 and 10, the height h cm of children can be modeled by
h=6a+80
where a is their age in years.
Extrapolation would be using this model for a 40 year old, which would give a prediction of h=6⋅40+80=320cm (around 10'6"). This is obviously crazy, and in this case it's obvious why a model for children's height does not apply to adults. But sometimes it's less obvious, and you should always be careful when extrapolating.
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