Answer
(a) opens down.
(b) vertex $(1,-3)$.
(c) $x=1$.
(d) y-intercept $f(0)=-5$.
(e) See graph.
(f) domain $(-\infty,\infty)$ range $(-\infty,-3]$
(g) increasing $(-\infty,1)$, decreasing $(1,\infty)$.
Work Step by Step
(a) From the given function $g(x)=-2x^2+4x-5$ with $a=-2\lt0$, we can determine the graph opens down.
(b) $g(x)=-2x^2+4x-5=-2(x^2-2x+1)-3=-2(x-1)^2-3$ and we can determine the vertex $(1,-3)$.
(c) We can determine the axis of symmetry $x=1$.
(d) $-2x^2+4x-5=0\Longrightarrow b^2-4ac=4^2-4(-2)(-5)=-24\lt0$ and we can determine the x-intercepts $none$, y-intercept $f(0)=-5$.
(e) See graph.
(f) Based on the graph, we can determine the domain $(-\infty,\infty)$ range $(-\infty,-3]$
(g) Based on the graph, we can determine the function is increasing $(-\infty,1)$, decreasing $(1,\infty)$.
