Linear Polynomials in the Wild

On the previous page you met four polynomial families. Of these, the linear family — degree 1 — is by far the most common in everyday life. A linear polynomial in $x$ is anything of the form

$ax + b$

where $a$ and $b$ are numbers and $a \neq 0$ (otherwise it would just be the constant $b$, not a linear polynomial).

• $a$ is the leading coefficient (the number multiplying $x$).
• $b$ is the constant term (the number standing alone).
• The variable can be any letter — $x$, $y$, $z$, $m$, $n$, $t$, … the name doesn't change anything.

Familiar linear polynomials:

• $4x$ (perimeter of a square of side $x$) — here $a = 4$, $b = 0$.
• $2x + 3$ (the function-machine example) — $a = 2$, $b = 3$.
• $200 + 50m$ (chess-club fee for $m$ matches) — $a = 50$, $b = 200$.
• $4z - 3$ — $a = 4$, $b = -3$.

Both Examples 4 (square perimeter) and 5 (chess fee) shared one striking feature: the values jumped by a constant amount each time the variable went up by 1. This is no coincidence. It's the defining property of a linear polynomial.

The rule. For any linear polynomial $f(n) = an + b$:

$f(n+1) - f(n) = [a(n+1) + b] - [an + b] = a$

The successive differences are always equal to the leading coefficient $a$. Always. No matter where you are on the line.

This gives a test for linearity:

• Compute $f(1), f(2), f(3), f(4), \ldots$
• Look at the differences $f(2) - f(1)$, $f(3) - f(2)$, etc.
• If all differences are equal → the pattern is linear.
• If the differences themselves change (like for $f(n) = n^2$, where they go $3, 5, 7, 9, \ldots$) → not linear.

The simulator above lets you toggle a comparison with $n^2$ to see this difference visually.