Linear function y = ax + b: all you need to know

The form

$f(x) = ax + b$

• $a$ = slope (steepness)
• $b$ = y-intercept (value of $f(0)$)

Its graph is a line.

Special cases

• $a = 0$: $f(x) = b$ (constant, horizontal line)
• $b = 0$: $f(x) = ax$ (proportional, line through origin)

Reading the slope

• $a > 0$: increasing (line goes up)
• $a < 0$: decreasing (line goes down)
• Large $|a|$: steep
• Small $|a|$: gentle

Compute $a$ from two points

If $A(x_A, y_A)$ and $B(x_B, y_B)$ are on the line:

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

“Rise over run.”

Compute $b$

$b = f(0)$. Or from $a$ and a point $(x_0, y_0)$: $b = y_0 - a x_0$

Example

Line through $A(1, 3)$ and $B(4, 9)$.

$a = 6/3 = 2$, $b = 3 - 2 = 1$. Equation: $y = 2x + 1$.

x-intercept

$f(x) = 0 \iff x = -b/a$ (if $a \neq 0$).

Common mistakes

1. Mixing up $a$ and $b$.
2. Flipping numerator and denominator in slope: it’s $\Delta y / \Delta x$.
3. Sign error in $y_0 - a x_0$.

?À toi de jouer

The line y = −3x + 5:

A.Decreasing, y-intercept 5B.Increasing, y-intercept −3C.Passes through (0, −3)D.Is horizontal

Valider