To determine the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex] in the function [tex]\( g(x) = (x-h)^2 + k \)[/tex], we need to compare it with the parent function [tex]\( f(x) = x^2 \)[/tex] and identify the transformations that have been applied.
### Step-by-Step Solution:
1. Identify the parent function:
- The given parent function is [tex]\( f(x) = x^2 \)[/tex].
2. Compare the given function with the parent function:
- The given function is [tex]\( g(x) = (x-h)^2 + k \)[/tex].
3. Identify the transformation components:
- By comparing [tex]\( g(x) = (x-h)^2 + k \)[/tex] with the parent function [tex]\( f(x) = x^2 \)[/tex], we note the following:
- The term [tex]\( (x-h)^2 \)[/tex] indicates a horizontal translation of [tex]\( h \)[/tex] units.
- The term [tex]\( +k \)[/tex] indicates a vertical translation of [tex]\( k \)[/tex] units.
4. Determine the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex]:
- In this transformation, [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are the values representing the horizontal and vertical shifts, respectively.
Therefore, the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are:
[tex]\[ h = h \][/tex]
[tex]\[ k = k \][/tex]
So, in the function
[tex]\[ g(x) = \left( x - \boxed{h} \right)^2 + \boxed{k}, \][/tex]
the boxed values [tex]\( h \)[/tex] and [tex]\( k \)[/tex] represent the horizontal and vertical translations of the parent function [tex]\( f(x) = x^2 \)[/tex], respectively.
