To find the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex] in the function [tex]\( g(x) = (x - h)^2 + k \)[/tex], we need to look at the vertex of the function.
The vertex form of a quadratic function [tex]\( g(x) = (x - h)^2 + k \)[/tex] tells us that the coordinates [tex]\( (h, k) \)[/tex] represent the vertex of the parabola.
In this problem, we're given that the vertex of the function [tex]\( g(x) \)[/tex] is at the point [tex]\((9, -8)\)[/tex].
Therefore:
- The value of [tex]\( h \)[/tex] is the x-coordinate of the vertex, which is [tex]\( 9 \)[/tex].
- The value of [tex]\( k \)[/tex] is the y-coordinate of the vertex, which is [tex]\( -8 \)[/tex].
So, we have:
[tex]\[ h = 9 \][/tex]
[tex]\[ k = -8 \][/tex]
Hence, the function [tex]\( g(x) = (x - h)^2 + k \)[/tex] can be written by substituting [tex]\( h \)[/tex] and [tex]\( k \)[/tex] with these values:
[tex]\[ g(x) = (x - 9)^2 - 8 \][/tex]
So, the filled-in function is:
[tex]\[ g(x) = (x - 9)^2 - 8 \][/tex]
The blanks in the function [tex]\( g(x) = (x - \square )^2 + \square \)[/tex] are filled as follows:
[tex]\[ g(x) = (x - \boxed{9})^2 + \boxed{-8} \][/tex]
Thus, the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are:
[tex]\[ h = 9 \][/tex]
[tex]\[ k = -8 \][/tex]
