Linear function

A linear function is a function given by

where $a$ and $b$ are fixed numbers. Its graph is — always, without exception — a straight line. The number $a$ is the slope, the number $b$ the intercept.

What a and b mean

Both numbers have a plain meaning, and the graph shows it at once:

• $b$ is the starting value: $f(0) = b$, the height at which the line crosses the $y$-axis;
• $a$ is the rate of change: by that much the value grows when the argument grows by one.

The dashed guide on the drawing is exactly how the slope is read off: one right, two up. Hence the other, equally good definition of $a$ — the rise divided by the run:

Monotonicity

Whether the function increases is decided by the sign of $a$ alone:

• $a > 0$ — the function increases, the line rises to the right;
• $a < 0$ — the function decreases, the line falls;
• $a = 0$ — the function is constant, $f(x) = b$, a horizontal line.

The intercept $b$ has nothing to do with monotonicity — it slides the line up or down without changing how steep it is.

The zero

The zero is the argument whose value is zero, that is the solution of $ax + b = 0$:

When $a = 0$ there is no point in using the formula: a constant function $f(x) = b$ either has no zeros at all (for $b \neq 0$) or takes the value zero everywhere (for $b = 0$).

The line through two points

Two points determine exactly one line, so from them the formula can be recovered. First the slope, then the intercept:

The order of subtraction in the numerator and the denominator must be the same — otherwise the sign comes out reversed. If the numerator starts from $y_2$, the denominator starts from $x_2$.

The general form

The same set of points can be described by an equation in which $y$ does not stand alone on the left:

That is the general form. To read the slope off it, just solve for $y$:

The change is only possible when $B \neq 0$. The equation $x = 3$ describes a vertical line, and such a line — as we know from the vertical line test — is not the graph of any function.

Parallel and perpendicular lines

Only the coefficient $a$ carries the steepness of a line, so both conditions on the mutual position of two lines are about it:

The condition gives a handy formula: a line perpendicular to one of slope $a$ has slope

the reciprocal with the opposite sign. For $a = 3$ that is $-\frac{1}{3}$; for $a = -\frac{2}{5}$ it is $\frac{5}{2}$.

Direct proportionality

When $b = 0$ the formula collapses to $y = ax$ and the graph passes through the origin. The ratio $\frac{y}{x}$ is then constant and equal to $a$ — this is direct proportionality.

Distance at a constant speed behaves like that ($s = vt$), and so does the cost of buying at a fixed price per item. When $b \neq 0$ a "joining fee" appears: a subscription added to a bill, the flag-fall of a taxi, the standing charge on an electricity invoice.

Exercises

One session covers everything this lesson teaches: the zero, the slope from two points, the whole formula from two points, the general form turned into the slope–intercept one, and the slope of a perpendicular line. Type a formula in any correct spelling — 2x+1, 1+2x and 2*x+1 all mean the same thing.

Common mistakes

• Swapping $a$ and $b$ — the steepness is $a$, the crossing with the $y$-axis is $b$.
• The sign of the zero — the formula gives $x_0 = -\frac{b}{a}$, not $\frac{b}{a}$.
• Subtracting in opposite orders — numerator and denominator must start from the same point.
• Dividing only one term — when solving for $y$, the whole side of the equation is divided.
• Perpendicularity as a bare reciprocal — the sign has to change too: $a_\perp = -\frac{1}{a}$.
• Calling a vertical line a linear function — $x = 3$ is not the graph of a function.