Answer
See graph,
vertex $(-2,11)$, axis $x=-2$,
x-intercepts $(-2\pm\frac{\sqrt {33}}{3},0)$, y-intercept $(0,-1)$,
domain $(-\infty,\infty)$, range $(-\infty,11]$, increasing $(-\infty,-2)$, decreasing $(-2,\infty)$.
Work Step by Step
Step 1. Graph the function $f(x)=-3x^2-12x-1=-3(x+2)^2+11$ as shown in the figure.
Step 2. We can identify the vertex $(-2,11)$, axis $x=-2$,
Step 3. For x-intercepts, let $f(x)=0$, we have $-3(x+2)^2+11=0$ thus $x=-2\pm\frac{\sqrt {33}}{3}$ or $(-2\pm\frac{\sqrt {33}}{3},0)$. For y-intercept, we have $f(0)=-1$ or $(0,-1)$,
Step 4. We can find domain $(-\infty,\infty)$, range $(-\infty,11]$, and largest open intervals of the domain over which each function is increasing $(-\infty,-2)$, decreasing $(-2,\infty)$.