SOLUTION: Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find that value. f(x) = -x2 - 3x - 9

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find that value. f(x) = -x2 - 3x - 9       Log On


   

Question 98523: Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find that value.
f(x) = -x2 - 3x - 9
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
The function has a maximum since the first term -x%5E2 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%2F%282a%29

From the equation y=-x%5E2-3x-9 we can see that a=-1 and b=-3

x=%28--3%29%2F%282%2A-1%29 Plug in b=-3 and a=-1


x=3%2F%282%2A-1%29 Negate -3 to get 3


x=%283%29%2F-2 Multiply 2 and -1 to get -2



x=-3%2F2 Reduce


So the axis of symmetry is x=-3%2F2


So the x-coordinate of the vertex is x=-3%2F2 (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%28-1.5%29

f%28x%29=-x%5E2-3x-9 Start with the given polynomial


f%28-1.5%29=-%28-1.5%29%5E2-3%28-1.5%29-9 Plug in x=-1.5


f%28-1.5%29=-%282.25%29-3%28-1.5%29-9 Raise -1.5 to the second power to get 2.25


f%28-1.5%29=-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%5E2-3x-9 you get


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