To determine the correct translated function [tex]\( g(x) \)[/tex] given the function [tex]\( f(x) = x^2 \)[/tex], we will analyze each provided option. Let's review each option step-by-step to see which transformation matches the desired translation.
1. Option A: [tex]\( g(x) = (x-4)^2 + 6 \)[/tex]
- Horizontal Shift: The term [tex]\( (x-4) \)[/tex] indicates a shift to the right by 4 units.
- Vertical Shift: The term [tex]\( +6 \)[/tex] indicates a shift upward by 6 units.
Thus, this transformation represents a right shift of 4 units and an upward shift of 6 units for the original function [tex]\( f(x) = x^2 \)[/tex].
2. Option B: [tex]\( g(x) = (x+6)^2 - 4 \)[/tex]
- Horizontal Shift: The term [tex]\( (x+6) \)[/tex] indicates a shift to the left by 6 units.
- Vertical Shift: The term [tex]\( -4 \)[/tex] indicates a shift downward by 4 units.
So, this transformation shifts the original function left by 6 units and downward by 4 units.
3. Option C: [tex]\( g(x) = (x-6)^2 - 4 \)[/tex]
- Horizontal Shift: The term [tex]\( (x-6) \)[/tex] indicates a shift to the right by 6 units.
- Vertical Shift: The term [tex]\( -4 \)[/tex] indicates a shift downward by 4 units.
This transformation shifts the original function right by 6 units and downward by 4 units.
4. Option D: [tex]\( g(x) = (x+4)^2 + 6 \)[/tex]
- Horizontal Shift: The term [tex]\( (x+4) \)[/tex] indicates a shift to the left by 4 units.
- Vertical Shift: The term [tex]\( +6 \)[/tex] indicates a shift upward by 6 units.
This transformation shifts the original function left by 4 units and upward by 6 units.
Based on the analysis:
- The translated function [tex]\( g(x) = (x-4)^2 + 6 \)[/tex] clearly matches a right shift of 4 units and an upward shift of 6 units.
Therefore, the correct equation of the translated function [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = (x-4)^2 + 6 \][/tex]
