$f(x)$ is just $y$ in disguise — the bracket tells you what replaces $x$, no "sub $x=$" needed. Quadratic expression inside means quadratic function. Reflex: Whatever sits in the bracket replaces every $x$.

Only works directly when coefficient of $x^2$ is 1. If not, factorize it out, complete the square inside, then multiply back. Reflex: Coefficient not 1? Factor out first — multiply back into BOTH terms.

$a>0$: minimum at $(h,k)$; $a<0$: maximum at $(h,k)$. The max/min value of $y$ is $k$. Reflex: $(x+3)^2$ means $h=-3$, not $+3$.

The curve opens upwards with its lowest point above the $x$-axis — that needs $a>0$ and $k>0$. Reflex: Two green lights — $a$ positive AND $k$ positive, miss either and it's not always positive.

"Initial" → sub $t=0$. "Reaches" a height → set expression equal, solve (use formula if it won't factorize). "Greatest/least" → complete the square, read turning point. Reflex: Let the question word pick your tool — initial, reaches, or greatest.