To determine the translation required to obtain the function [tex]\( g(x) \)[/tex] from the original function [tex]\( f(x) = x^2 \)[/tex], we need to analyze how the function [tex]\( f(x) \)[/tex] has been shifted both horizontally and vertically.
Consider the general form of a translated quadratic function:
[tex]\[ g(x) = (x - h)^2 + k \][/tex]
Here, [tex]\( h \)[/tex] represents the horizontal shift and [tex]\( k \)[/tex] represents the vertical shift:
- If [tex]\( h > 0 \)[/tex], the graph is shifted to the right by [tex]\( h \)[/tex] units.
- If [tex]\( h < 0 \)[/tex], the graph is shifted to the left by [tex]\( |h| \)[/tex] units.
- If [tex]\( k > 0 \)[/tex], the graph is shifted upward by [tex]\( k \)[/tex] units.
- If [tex]\( k < 0 \)[/tex], the graph is shifted downward by [tex]\( |k| \)[/tex] units.
Let's examine each given option one by one:
1. [tex]\( g(x) = (x - 4)^2 + 6 \)[/tex]
- This represents a horizontal shift to the right by 4 units (since [tex]\( h = 4 \)[/tex]).
- Also, it represents a vertical shift upward by 6 units (since [tex]\( k = 6 \)[/tex]).
2. [tex]\( g(x) = (x + 6)^2 - 4 \)[/tex]
- This represents a horizontal shift to the left by 6 units (since [tex]\( h = -6 \)[/tex]).
- Also, it represents a vertical shift downward by 4 units (since [tex]\( k = -4 \)[/tex]).
3. [tex]\( g(x) = (x - 6)^2 - 4 \)[/tex]
- This represents a horizontal shift to the right by 6 units (since [tex]\( h = 6 \)[/tex]).
- Also, it represents a vertical shift downward by 4 units (since [tex]\( k = -4 \)[/tex]).
4. [tex]\( g(x) = (x + 4)^2 + 6 \)[/tex]
- This represents a horizontal shift to the left by 4 units (since [tex]\( h = -4 \)[/tex]).
- Also, it represents a vertical shift upward by 6 units (since [tex]\( k = 6 \)[/tex]).
Given the translation described in the question, the correct equation of the translated function [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = (x - 4)^2 + 6 \][/tex]
Thus, the correct option is:
[tex]\[ \boxed{1} \][/tex]
