Answer
See graph:
(a) $(-3,4)$.
(b) $x=-3$.
(c) $(-\infty,\infty)$.
(d) $(-\infty,4]$.
(e) $(-\infty,-3)$.
( f ) $(-3,\infty)$.
Work Step by Step
See graph:
(a) Given $f(x)=-\frac{1}{2}x^2-3x-\frac{1}{2}=-\frac{1}{2}(x^2+6x+9)+4=-\frac{1}{2}(x+3)^2+4$, we can find its vertex at $(-3,4)$.
(b) The axis can be found as $x=-3$.
(c) The domain can be found as $(-\infty,\infty)$.
(d) The range can be found as $(-\infty,4]$.
(e) the largest open interval of the domain over which the function is increasing can be identified as $(-\infty,-3)$.
( f ) the largest open interval over which the function is decreasing can be identified as $(-3,\infty)$.