Thank you for the opportunity to help you with your question!

a) f(x) = -2x^2 + 18x, so comparing f(x) to ax^2 + bx + c, we have a = -2, b = 18, c = 0. The formula for the vertex is (-b/2a) which in this case means x = -18/2(-2) = 18/4 = 9/2 = 4.5. Because a < 0, the parabola opens downward, so at the vertex, f(x) is at its maximum.

b) Factoring f(x) and setting it equal to zero, we get -2x(x - 9) = 0. This is a true statement only if x = 0 or x = 9.

c) I'm having troubel with my plotter. The graph should look an open umbrella, kind of like an upside down U. It will cross the x-axis at x = 0 and x = 9 with a y-coordinate of zero at each of these two points.