Question 1130541

 {{{f(x) = -x^2 -6x -7}}}

here you have parabola that opens downward; so, it has a maximum at vertex

range will be all below {{{y}}} coordinate of the vertex

write your equation in vertex form:

 {{{f(x) = -(x^2 +6x) -7}}}

 {{{f(x) = -1*(x^2 +6x+b^2)-1*(-b^2) -7}}}

 {{{f(x) = -(x^2 +6x+3^2)+3^2 -7}}}

 {{{f(x) = -(x+3)^2)+9 -7}}}

{{{f(x) = -(x+3)^2)+2}}}=> {{{k=2}}} =>{{{y}}} coordinate of the vertex

then, range is:

{  {{{f(x)}}} element {{{R}}} : {{{f(x) <=2}}} }


{{{ graph( 600, 600, -10, 10, -10, 10, -(x+3)^2+2) }}}