Functions

$k>0$ shifts UP by $k$. Intuitive direction.

$h>0$ shifts RIGHT by $h$ (counter-intuitive: SUBTRACTING $h$ moves right). The vector form is $\binom{h}{k}$.

Stretches by factor $a$ from $x$-axis. $a>1$ ⇒ further from x-axis. $0<a<1$ ⇒ closer (compression).

Stretches by factor $1/b$ from $y$-axis (reciprocal!). $b>1$ ⇒ COMPRESSION; $0<b<1$ ⇒ STRETCH.

$a$ vert stretch · $b$ horiz stretch · $h$ horiz shift · $k$ vert shift. Order matters when mixing inside and outside.

$y=|f(x)|$ reflects parts BELOW the $x$-axis ABOVE it. $y=f(|x|)$ replaces the $x<0$ part with a mirror image of the $x>0$ part across the $y$-axis.

Zeros of $f$ ⇒ vertical asymptotes of $1/f$. Where $f$ has a min, $1/f$ has a max (at the same $x$). $1/f$ has the same sign as $f$.

VA at $x=-d/c$; HA at $y=a/c$. Equivalent transformed form $y=\tfrac{A}{B(x-h)}+k$.

Odd, axes are asymptotes, self-inverse. $D = R = \mathbb{R}\setminus\{0\}$.

Domain: $ax+b\ge 0$. Range: $y\ge 0$ (or $\le 0$ if outer minus). Curve starts at boundary and rises with decreasing slope.

Solve both cases and CHECK each against the original equation.

Set denominator $=0$; the values are VAs (unless they cancel with the numerator).

Compare degrees: numerator deg $<$ denominator deg ⇒ $y=0$. Equal degrees ⇒ $y=$ leading-coeff ratio. Higher numerator ⇒ no HA (slant or beyond).

Apply inside-the-bracket transformations first (in reverse algebra order): translate inside ⇒ then stretch inside. Outside transformations follow in natural order.

$f(x-h)$ shifts RIGHT by $h$, $f(x+h)$ shifts LEFT by $h$. Inside the bracket the sign flips its apparent meaning.

If $B<0$ there is no solution. If $B$ contains $x$, always check — squaring or splitting can introduce false roots.