Linear Functions and Constant Rate of Change

Definition A linear function is a function of the form

where $m$ and $b$ are real constants. Its domain is $\mathbb{R}$ unless a smaller domain is specified. On the domain $\mathbb{R}$ its graph is a nonvertical straight line, and every nonvertical line is the graph of exactly one such function. When $m = 0$ the function is constant. Some books call $mx + b$ an affine function and keep the word linear for the case $b = 0$, but in precalculus linear means any function of this form.

Definition The slope of the linear function $f(x) = mx + b$ is the number $m$. For any two inputs $x_1 \ne x_2$,

so the slope can be computed from any two distinct points of the line. A positive slope means the line rises from left to right, a negative slope means it falls, and slope $0$ means it is horizontal.

Definition A function $f$ defined on an interval $I$ has constant rate of change $m$ on $I$ if

for every pair of distinct inputs $x_1$ and $x_2$ in $I$. This happens exactly when $f(x) = mx + b$ on $I$ for some constant $b$. Indeed, fixing one input $x_1$ and setting $b = f(x_1) - mx_1$, the condition says $f(x) - f(x_1) = m(x - x_1)$, that is $f(x) = mx + b$, for every $x$ in $I$. In particular, equal steps in the input always produce equal steps in the output.

Definition The slope-intercept form of a nonvertical line is

where $m$ is the slope and $(0, b)$ is the point where the line crosses the $y$-axis. Every nonvertical line has exactly one equation of this form.

Definition The line with slope $m$ through the point $(x_1, y_1)$ has the equation

A point $(x, y)$ with $x \ne x_1$ satisfies this equation exactly when $\frac{y - y_1}{x - x_1} = m$, which says that the slope from $(x_1, y_1)$ to $(x, y)$ is $m$. The point $(x_1, y_1)$ itself satisfies it too.

Definition The $y$-intercept of the graph of $f$ is the point $(0, f(0))$, which exists when $0$ is in the domain. An $x$-intercept is a point $(a, 0)$ with $f(a) = 0$, and the number $a$ is called a zero of $f$. On the domain $\mathbb{R}$, a linear function $f(x) = mx + b$ has $y$-intercept $(0, b)$. If $m \ne 0$ it has exactly one $x$-intercept,

If $m = 0$ and $b \ne 0$ it has none, and if $m = 0$ and $b = 0$ its graph is the whole $x$-axis. Many books also call the numbers $b$ and $-\frac{b}{m}$ themselves the intercepts.

Definition A quantity $y$ is directly proportional to a quantity $x$ if there is a constant $k \ne 0$ with

for all values considered. The number $k$ is the constant of proportionality, and for $x \ne 0$ it equals the ratio $\frac{y}{x}$. The graph lies on the line through the origin with slope $k$. A linear function $y = mx + b$ with $b \ne 0$ does not make $y$ proportional to $x$. Its line passes through $(0, b)$ instead of the origin, and for $x \ne 0$ the ratio $\frac{y}{x} = m + \frac{b}{x}$ changes with $x$.

Definition If the input of a linear function is measured in units $U$ and the output in units $V$, its slope is measured in $V$ per $U$, because the slope is a change in output divided by a change in input. The constant $b$ has the units of the output, and so does each term of $mx + b$.

Definition The horizontal line through $(a, c)$ has the equation $y = c$. It is the graph of the constant function $f(x) = c$ and has slope $0$. The vertical line through $(a, c)$ has the equation $x = a$ and consists of all points $(a, y)$ with $y$ real. It is not the graph of any function $y = f(x)$, because it pairs the single input $a$ with infinitely many outputs, while a function gives each input exactly one output. In other words, it fails the vertical line test. Its slope is undefined, since any two of its points have $x_2 - x_1 = 0$.