1.35 Linear Function and Model Definitions

Linear Function and Model Definitions establishes the vocabulary for functions whose graphs form straight lines, describing the general symbolic form such functions take, the application of that form to represent real-world situations, the specific starting value such a model begins from, and the process of using the model to estimate values beyond the ones directly observed.

Linear Function

A linear function is a function whose graph is a straight line, expressible in the form f(x) = mx + b, where m represents the constant slope of the line and b represents the value of the function when the input is zero. Every linear function has a constant rate of change, meaning equal increases in the input always produce equal increases (or decreases) in the output, regardless of where along the domain those increases occur.

Linear Model

A linear model is the application of a linear function to represent a real-world relationship between two quantities, with the slope m and the constant b assigned specific contextual meaning drawn from the situation being described, rather than remaining abstract symbols. A linear model describing the cost of a service might assign m the meaning of a per-unit charge and assign b the meaning of a flat starting fee, tying the abstract linear function directly to the quantities in the described scenario.

Linear Initial Value

The linear initial value is the value of a linear model's output when its input is zero, corresponding to the constant b in the form f(x) = mx + b, and typically representing a starting amount, fixed fee, or baseline condition present in the modeled situation before any change described by the rate of change has occurred. In a model of total cost based on a flat fee plus a per-mile charge, the linear initial value is the flat fee itself, the amount owed even before any miles have been driven.

Linear Prediction

Linear prediction is the process of using a linear model to estimate an output value for an input that was not directly observed or measured, by substituting that input into the model's function rule and evaluating the result. Once a linear model has been established from known data, linear prediction extends its usefulness beyond the original observations, estimating outcomes for new input values under the assumption that the same constant rate of change continues to apply.

Together, these definitions describe how the abstract linear function is grounded into a linear model tied to a real situation, with the linear initial value anchoring the model's starting point and linear prediction extending the model to estimate outcomes for inputs beyond those originally observed.