Question 133089
Vertex:


To find the vertex, we first need to find the axis of symmetry (ie the x-coordinate of the vertex)

To find the axis of symmetry, use this formula:


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


From the equation {{{y=-x^2-2x+8}}} we can see that a=-1 and b=-2


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



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



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




{{{x=-1}}} Reduce



So the axis of symmetry is  {{{x=-1}}}



So the x-coordinate of the vertex is {{{x=-1}}}. Lets plug this into the equation to find the y-coordinate of the vertex.



Lets evaluate {{{f(-1)}}}


{{{f(x)=-x^2-2x+8}}} Start with the given polynomial



{{{f(-1)=-(-1)^2-2(-1)+8}}} Plug in {{{x=-1}}}



{{{f(-1)=-(1)-2(-1)+8}}} Raise -1 to the second power to get 1



{{{f(-1)=-(1)--2+8}}} Multiply 2 by -1 to get -2



{{{f(-1)=-1+2+8}}} Negate any negatives



{{{f(-1)=9}}} Now combine like terms



So the vertex is (-1,9)




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Intercepts:


Y-intercept:


To find the y-intercept, simply plug in x=0 and simplify




{{{f(x)=-1x^2-2x+8}}} Start with the given function



{{{f(0)=-1(0)^2-2(0)+8}}} Plug in {{{x=0}}}



{{{f(0)=-1*0-2*0+8}}} Raise 0 to the 2nd power to get 0



{{{f(0)=0-2*0+8}}} Multiply -1 and 0 to get 0



{{{f(0)=0-0+8}}} Multiply 2 and 0 to get 0



{{{f(0)=0+8}}} Subtract 0 from 0 to get 0



{{{f(0)=8}}} Add 0 and 8 to get 8



So when {{{x=0}}}, we have {{{y=8}}} which means that the y-intercept is (0,8)





X-intercept:


To find the x-intercept, plug in y=0 and solve for x



{{{-x^2-2x+8=0}}} Start with the given equation


{{{(-x-4)(x-2)=0}}} Factor the left side (note: if you need help with factoring, check out this <a href=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>solver</a>)




Now set each factor equal to zero:

{{{-x-4=0}}} or  {{{x-2=0}}} 


{{{x=-4}}} or  {{{x=2}}}    Now solve for x in each case



So our answer is 

 {{{x=-4}}} or  {{{x=2}}} 



which means that the x-intercepts are 


(-4,0) and (2,0)




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Notice if we graph {{{y=-x^2-2x+8}}}, we can visually verify our answer



{{{ graph( 500, 500,-10, 10, -10, 10, -x^2-2x+8) }}}