Linear function: complete lesson (graph, slope, y-intercept)

Definition

A linear function (affine function) has the form:

$f(x) = a x + b$

with $a$ and $b$ fixed real numbers.

• $a$ is the slope.
• $b$ is the y-intercept (value at $0$: $f(0) = b$).

Special cases

• If $b = 0$: strict linear function $f(x) = a x$ (line through origin).
• If $a = 0$: constant function $f(x) = b$ (horizontal line).

Graph

The graph of a linear function is a line:

• Not vertical (since $x$ is free).
• Passes through $(0, b)$.
• Passes through $(1, a + b)$.

Compute the slope

Given two points $A(x_A, y_A)$ and $B(x_B, y_B)$ on the line:

$a = \dfrac{y_B - y_A}{x_B - x_A}$

Careful: $x_A \neq x_B$, otherwise the line is vertical (not a function).

Monotonicity

• $a > 0$ ⇒ $f$ strictly increasing on $\mathbb{R}$.
• $a < 0$ ⇒ $f$ strictly decreasing on $\mathbb{R}$.
• $a = 0$ ⇒ $f$ constant.

Find the expression from two points

Step 1: compute $a$ with the formula above. Step 2: find $b$ by writing that $A$ lies on the line: $y_A = a x_A + b$.

Example: line through $A(1, 5)$ and $B(3, 11)$.

$a = (11 - 5) / (3 - 1) = 3$.

$5 = 3 \times 1 + b$ ⇒ $b = 2$.

So $f(x) = 3 x + 2$.

Sign of a linear function

If $a \neq 0$, $f$ vanishes at $x_0 = -b/a$.

Sign:

• $a > 0$: negative before $x_0$, positive after.
• $a < 0$: positive before $x_0$, negative after.

“Sign of $a$ after the root, opposite sign before”.

Two lines: parallel, intersecting, coincident

Let $d : y = a x + b$ and $d' : y = a' x + b'$.

• $a = a'$ and $b \neq b'$ ⇒ strictly parallel.
• $a = a'$ and $b = b'$ ⇒ coincident.
• $a \neq a'$ ⇒ intersecting at a unique point.

Real-life applications

• Subscription: $f(x) = 0.5 \, x + 20$ ($20 fixed +$0.50/call).
• Production cost: $f(x) = 3 \, x + 100$ ($100 fixed +$3/unit).
• Temperature conversion: $T_F = 1.8 \, T_C + 32$.

Common mistakes

1. Confusing $a$ and $b$ in $f(x) = a x + b$.
2. Forgetting to divide in the slope formula.
3. Thinking a vertical line is a linear function (false: many $y$ for one $x$).
4. Thinking $f(x) = 3$ is not linear (it is, with $a = 0$).