Question 118377
To find the roots of {{{f(x)=x^2+6x-16}}}, set the entire function equal to zero

 {{{x^2+6x-16=0}}}



{{{(x+8)(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+8=0}}} or  {{{x-2=0}}} 


{{{x=-8}}} or  {{{x=2}}}    Now solve for x in each case



So the roots are


 {{{x=-8}}} or  {{{x=2}}} 




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Now let's find the vertex (ie the turning point)


To find the x coordinate of the vertex, simply use this formula {{{x=-b/2a}}}


Looking at {{{f(x)=x^2+6x-16}}}, we can see that {{{a=1}}} and {{{b=6}}}. 


{{{x=-6/2(1)}}}  Plug {{{a=1}}} and {{{b=6}}} into the formula. 


{{{x=-6/2}}}  Multiply


{{{x=-3}}} Reduce



So the x-coordinate of the vertex is  {{{x=-3}}} 



To find the y coordinate, simply plug in {{{x=-3}}} into  {{{f(x)=x^2+6x-16}}} to evaluate {{{f(-3)}}}





Let's evaluate {{{f(-3)}}}



{{{f(x)=x^2+6x-16}}} Start with the given function.



{{{f(-3)=(-3)^2+6(-3)-16}}} Plug in {{{x=-3}}}. In other words, replace each x with -3.



{{{f(-3)=(9)+6(-3)-16}}} Evaluate {{{(-3)^2}}} to get 9.



{{{f(-3)=(9)+-18-16}}} Multiply 6 and -3 to get  -18



{{{f(-3)=-25}}} Now combine like terms


So the vertex is: (-3,-25)




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Summary:


So our roots are 


{{{x=-8}}} or {{{x=2}}}


and our vertex is at the point 


(-3,-25)




Now let's graph these three points


{{{drawing(500,500,-15,15,-30,5, 
graph(500,500,-15,15,-30,5,0),
grid(1),
circle(-8,0,0.12),
circle(-8,0,0.15),
circle(-8,0,0.18),

circle(2,0,0.12),
circle(2,0,0.15),
circle(2,0,0.18),


circle(-3,-25,0.12),
circle(-3,-25,0.15),
circle(-3,-25,0.18)

) }}}



Now connect the points with a parabola (note: the more points you plot the more accurate the graph is)


{{{drawing(500,500,-15,15,-30,5, 
graph(500,500,-15,15,-30,5,0,x^2+6x-16),
grid(1),
circle(-8,0,0.12),
circle(-8,0,0.15),
circle(-8,0,0.18),

circle(2,0,0.12),
circle(2,0,0.15),
circle(2,0,0.18),


circle(-3,-25,0.12),
circle(-3,-25,0.15),
circle(-3,-25,0.18)

) }}}