Suppose $f$ is a real valued function with domain a subset of  real numbers, $D$.

Definition: $f$  is a linear function if there a numbers $m$ and $b$ so that for any $x \in D$,  $f(x) = mx +b$
Comment: The letters $m$ and $b$ have been used to represent for these numbers for hundreds of years, but there are alternatives. Here are two alternatives:
i. $f$  is a linear function if there a numbers $a_1$ and $a_0$ so that for any $x \in D$,  $f(x) = a_1 x +a_0$.
ii. $f$  is a linear function if there a numbers $A$ and $B$ so that for any $x \in D$,  $f(x) = Ax +B$.
The number $m$ is sometimes referred to as "the slope," "the rate," or the "linear coefficient " of the linear function.
The number $b$ is sometimes referred to as "the $y$- intercept," "the initial value," or the "constant " of the linear function.