Lesson 3 — Affine functions (degree 1)

Definition and properties

• $a$: slope coefficient (or angle of inclination)
• $b$: vertical intercept (y-intercept)
• Graph: a straight line. $a > 0$: increasing. $a < 0$: decreasing. $a = 0$: constant.

$$\frac{f(x_2) - f(x_1)}{x_2 - x_1} = a$$

Read

(1)

"The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is $m = (y_2 - y_1)/(x_2 - x_1)$." — OpenStax College Algebra 2e, §2.2

Zero of the function and intercepts

$f(x) = 0 \iff x = -b/a$ (when $a \neq 0$). The pair $(0, b)$ is the vertical intercept. The pair $(-b/a, 0)$ is the zero (or horizontal intercept).

Uniqueness theorem for two points

Proof (sketch). Existence: define $a$ by the formula above and $b = y_1 - a x_1$. It is verified that $f(x_1) = y_1$ by construction, and $f(x_2) = a(x_2 - x_1) + y_1 = (y_2 - y_1) + y_1 = y_2$. Uniqueness: if $g(x) = a' x + b'$ also satisfies $g(x_i) = y_i$, then $a' = (y_2 - y_1)/(x_2 - x_1) = a$ and $b' = y_1 - a' x_1 = b$. ∎

Composition and operations

Let $f(x) = a_1 x + b_1$ and $g(x) = a_2 x + b_2$. Then:

• Sum: $(f + g)(x) = (a_1 + a_2) x + (b_1 + b_2)$ — affine, with slopes added.
• Composition: $(f \circ g)(x) = a_1 (a_2 x + b_2) + b_1 = a_1 a_2 x + (a_1 b_2 + b_1)$ — affine, with slopes multiplied.
• Inverse (if $a_1 \neq 0$): $f^{-1}(y) = (y - b_1)/a_1$ — also affine, with slope $1/a_1$.

The set of invertible affine functions ($a \neq 0$) under composition forms a group — the structure $(\mathbb{R}^* \times \mathbb{R}, \circ)$. This observation will be used in linear algebra (Lesson 31+) and affine geometry.

Family of parallel lines

Family of concurrent lines