Answer
See the sketch.
Work Step by Step
The domain of $f$ is (−∞,∞) and there is no symmetry. There are no asymptotes since $f$ is a polynomial. $f'(x) = 3x^2 − 12x − 135 = 3(x − 9)(x + 5)$, which is $0$ for $x = 9$ and $x = −5$. $f'(x) > 0$ on (−∞,−5), and on (9,∞), so $f$ is increasing on those intervals. $f'(x) < 0$ on (−5, 9), so $f$ is decreasing on that interval. There is a local maximum at $x = −5$ and a local minimum at $x = 9$.
$f''(x) = 6x−12$, which is $0$ for $x = 2$. $f''(x) > 0$ on (2,∞), so $f$ is concave up on that interval. $f''(x) < 0$ on (−∞, 2), so $f$ is concave down on that interval. There is a point of inflection at $x = 2$. The y-intercept is $0$ and the x-intercepts are $−9$ and $15$.
