Certainly! Let's break down the process step-by-step.
We are given a function [tex]\( g(x) = (x - h)^2 + k \)[/tex], which is in vertex form for a quadratic function. This form is derived from the parent function [tex]\( f(x) = x^2 \)[/tex], where the vertex is located at [tex]\( (h, k) \)[/tex].
In the problem, it is specified that the vertex of the function [tex]\( g(x) \)[/tex] is at the point (9, -8).
### Step-by-Step Solution:
1. Identify the Vertex Form:
The vertex form of a quadratic function is given by:
[tex]\[
g(x) = (x - h)^2 + k
\][/tex]
Here, [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are the coordinates of the vertex (h, k).
2. Substitute the Vertex Coordinates:
We are provided with the vertex [tex]\((9, -8)\)[/tex]. Therefore, [tex]\( h = 9 \)[/tex] and [tex]\( k = -8 \)[/tex].
3. Fill in the Values:
Replacing [tex]\( h \)[/tex] and [tex]\( k \)[/tex] in the vertex form equation, we have:
[tex]\[
g(x) = (x - 9)^2 - 8
\][/tex]
### Final Answer:
So, the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are:
[tex]\[
h = 9 \quad \text{and} \quad k = -8
\][/tex]
Substituting these values back into the function [tex]\( g(x) \)[/tex],
[tex]\[
g(x) = (x - 9)^2 - 8
\][/tex]
Therefore, the completed function is:
[tex]\[
g(x) = (x - 9)^2 - 8
\][/tex]
In conclusion, the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are:
[tex]\[
h = 9, \quad k = -8
\][/tex]
