As you are using # f(x) # I am assuming you are at a higher level of Mathematics.

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#color(brown)("Using Calculus with shortcuts")#

#color(blue)("To find "x_("vertex")#

Given: #f(x)=-3x^2-6x-7#..............(1)

#f'(x)=-6x-6#

Equating to zero gives #color(blue)(x_("vertex")=-1)#

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#color(blue)("To find "y_("vertex")#

Substitute #x=-1# into equation (1) giving

#y_("vertex")= -3(-1)^2-6(-1)-7#

#color(blue)(y_("vertex")= -4)#

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Vertex#->(x,y)->(-1,-4)#

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This is a maximum as #-3x^2# being negative is indicative of an inverted U shape. Also the 2nd differential is negative which is also indicative of a maximum

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#color(brown)("NOT using Calculus")#

#color(blue)("I am going to show you a really cool trick!")#

Write as:# y= (-3x^2-6x)-7#

Factor the -3 out

#y= -3(x^2+2x)-7#............................(1)

Now consider the #+2" from "2x# inside the bracket

Multiply this by #(-1/2)#

#(-1/2)xx(+2)=-1#

This is the value you are after for #x# so:

#color(blue)(x_("vertex")=-1)#

Substitute this x value into the original equation to find the value of #y_("vertex")#