Linear and Piecewise Linear Models

The linear model

A linear function has the form $f(x)=mx+c$, where $m$ is the gradient (slope) and $c$ is the $y$-intercept. Linear models are the workhorse of applied mathematics because so many real relationships are, at least approximately, constant-rate. The gradient $m$ is the change in output per one-unit change in input — dollars per kilometer, degrees per minute, meters per second. The intercept $c$ is the output when the input is zero — a fixed fee, a starting amount, or an initial value.

Worked example (taxi fare). A taxi charges a \$3.50 flag-down fee plus \$1.80 per kilometer. The fare is $F(d)=3.50+1.80d$ where $d$ is distance in km. The intercept \$3.50 is what you pay before moving; the gradient \$1.80 is the cost of each extra kilometer. A 6 km trip costs $F(6)=3.50+10.80=\