Question 98523
The function has a maximum since the first term {{{-x^2}}} has a negative coefficient. 



To find the max, you need to find the axis of symmetry first



To find the axis of symmetry, use this formula:


{{{x=-b/(2a)}}}


From the equation {{{y=-x^2-3x-9}}} we can see that a=-1 and b=-3


{{{x=(--3)/(2*-1)}}} Plug in b=-3 and a=-1



{{{x=3/(2*-1)}}} Negate -3 to get 3



{{{x=(3)/-2}}} Multiply 2 and -1 to get -2




{{{x=-3/2}}} Reduce



So the axis of symmetry is  {{{x=-3/2}}}



So the x-coordinate of the vertex is {{{x=-3/2}}} (which is {{{x=-1.5}}} in decimal form). Lets plug this into the equation to find the y-coordinate of the vertex.



Lets evaluate {{{f(-1.5)}}}


{{{f(x)=-x^2-3x-9}}} Start with the given polynomial



{{{f(-1.5)=-(-1.5)^2-3(-1.5)-9}}} Plug in {{{x=-1.5}}}



{{{f(-1.5)=-(2.25)-3(-1.5)-9}}} Raise -1.5 to the second power to get 2.25



{{{f(-1.5)=-6.75}}} Now combine like terms



So the vertex is (-1.5,-6.75)



So that means the functions highest value is -6.75 which means the functions maximum is -6.75



Notice if you graph the equation {{{y=-x^2-3x-9}}} you get

{{{drawing(900,900,-11.5,8.5,-16.75,3.25,
graph(900,900,-11.5,8.5,-16.75,3.25, -x^2-3x-9)
)}}}


Notice the highest y value is -6.75. So our answer is graphically verified.