ANSWER
[tex]h = - 4,k = - 6[/tex]
Minimum value: y=-6
Occurs at: x=-4
EXPLANATION
Given
[tex]f(x) = {x}^{2} + 8x + 10[/tex]
We need to write the equivalent of this function in vertex form:
[tex]f(x) = ({x - h)}^{2} + k[/tex]
where (h,k) is the vertex.
We must complete the square to get the function to this form.
We add and subtract the square of half the coefficient of x.
[tex]f(x) = {x}^{2} + 8x +( \frac{8}{2}) ^{2} - ( \frac{8}{2}) ^{2} + 10[/tex]
This gives us;
[tex]f(x) = {x}^{2} + 8x +16- 16 + 10[/tex]
The first three terms form a perfect square quadratic trinomial.
[tex]f(x) =( x + 4) ^{2} - 6[/tex]
or
[tex]f(x) =( x - - 4) ^{2} - 6[/tex]
Therefore we compare to
[tex]f(x) = ({x - h)}^{2} + k[/tex]
h=-4 and k=-6
The minimum value of the function is
[tex]y = - 6[/tex]
and this occurs at;
[tex]x = - 4[/tex]
