Answer
a. falls to the left and also falls to the right.
b. neither
c. See graph.
Work Step by Step
a. The leading term of the function
$f(x)=-x^4+6x^3-9x^2$
is $-x^4$, with a coefficient of $-1$ and an even power. Thus, we can identify its end behaviors as $x\to-\infty, y\to-\infty$ and $x\to\infty, y\to-\infty$. That is, the curve falls to the left and also falls to the right.
b. We test
$f(-x)=-(-x)^4+6(-x)^3-9(-x)^2=-x^4-6x^3-9x^2$
as $f(-x)\ne f(x)$ and $f(-x)\ne -f(x)$, the function is neither symmetric with respect to the y-axis nor with the origin.
c. See graph.
