Transformations and Function Model Selection

Reading g(x) = a·f(b(x + c)) + d

Outside the function: vertical. $a$ stretches by $|a|$ (reflects over the $x$-axis if $a<0$); $d$ shifts up. Inside the function: horizontal, and inverted. $b$ shrinks by a factor of $\frac{1}{|b|}$ (reflects over the $y$-axis if $b<0$); $+c$ shifts left. When the argument is written as $bx+k$, factor first: $f(2x+6)=f\left(2(x+3)\right)$ — shrink by $\tfrac12$, then shift left $3$, not left $6$.

Asymptotes ride along with the transformation: for a rational parent, a vertical asymptote $x=v$ moves to the solution of $b(x+c)=v$, and a horizontal asymptote $y=h$ moves to $y=ah+d$.

Choosing a model (1.13–1.14)

Over equally spaced inputs: constant first differences $\Rightarrow$ linear; constant $n$-th differences $\Rightarrow$ degree-$n$ polynomial. Data that levels off toward a ceiling calls for a rational model — a polynomial is disqualified because every nonconstant polynomial is unbounded: $\lim_{x\to\infty}p(x)=\pm\infty$.

Every model answer needs its "assumptions and limitations" sentence: the domain where the model makes sense, the meaning of its asymptote, and where it must not be extrapolated.