A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, or the concentration of a product in a chemical reaction.
The second stage is to apply mathematics as we know to the mathematical model that we have formulated to derive mathematical conclusions.
Linear Models
When we say that y is a linear function of x, we mean that the graph of the function is a line, so we can use the slope-intercept form of the equation of a line to write a formula for where m is the slope of the line and b is the y-intercept. A characteristic feature of a linear function is that they grow at a constant rate. We represent the linear function as y = f(x) = mx + b.
A better linear model is obtained by a procedure from statistics called linear regression. Using a graphing calculator we can enter data and return the slope and y-intercept of the regression line to determine our least squares model.
We use linear regression to extrapolate predicted values outside of the region of observations. However, these predictions are somewhat risky because it involves time remote from our observations.