Homographic function: (ax+b)/(cx+d) and its hyperbola

Definition

A homographic function has the form:

$f(x) = \dfrac{ax + b}{cx + d}$

with $c \neq 0$ and $ad - bc \neq 0$ (otherwise $f$ is constant).

Domain

Denominator must not vanish: $cx + d \neq 0$, so $x \neq -d/c$.

$D_f = \mathbb{R} \setminus \{-d/c\}$

Asymptotes

• Vertical asymptote: $x = -d/c$ (where the denominator vanishes).
• Horizontal asymptote: $y = a/c$ (limit at $\pm\infty$: quotient of dominant terms).

Derivative and monotonicity

$f'(x) = \dfrac{ad - bc}{(cx + d)^2}$

Denominator is always positive (square), so sign of $f'$ = sign of $ad - bc$.

• If $ad - bc > 0$ ⇒ $f$ increasing on each interval.
• If $ad - bc < 0$ ⇒ $f$ decreasing on each interval.

⚠️ On each interval separately, never on all of $D_f$.

Graph

A hyperbola whose vertical and horizontal asymptotes are the translated axes.

Center of symmetry: $\big(-d/c \, , \, a/c\big)$ = intersection of asymptotes.

Example: $f(x) = \dfrac{2x + 1}{x - 3}$

• Domain: $x \neq 3$.
• Vertical asymptote: $x = 3$.
• Horizontal asymptote: $y = 2$ (since $a/c = 2/1$).
• $ad - bc = 2 \times (-3) - 1 \times 1 = -7 < 0$ ⇒ $f$ decreasing on $(-\infty, 3)$ and on $(3, +\infty)$.

Common mistakes

1. Saying $f$ decreasing on $\mathbb{R} \setminus \{-d/c\}$ (false because of the break).
2. Forgetting the horizontal asymptote ($y = 0$ only when $a = 0$).
3. Confusing asymptote and tangent.
4. Computing $f(-d/c)$ (forbidden).

?À toi de jouer

Asymptotes of f(x) = (3x + 5)/(x − 2)?

A.x = 2 and y = 3B.x = −2 and y = 3C.x = 2 and y = 5D.x = 3 and y = 2

Valider