To determine the equation of the translated function [tex]\( g(x) \)[/tex] given [tex]\( f(x) = x^2 \)[/tex], let's analyze each provided option and compare them to standard transformation formulas.
### Step-by-Step Analysis:
#### Option 1: [tex]\( g(x) = (x-4)^2 + 6 \)[/tex]
1. Horizontal Translation: In [tex]\( (x-4)^2 \)[/tex], the [tex]\( x \)[/tex] is replaced by [tex]\( x-4 \)[/tex]. This translates the graph of [tex]\( f(x) = x^2 \)[/tex] horizontally 4 units to the right.
2. Vertical Translation: The [tex]\( +6 \)[/tex] outside the squared term represents an upward translation by 6 units.
So, [tex]\( g(x) \)[/tex] in this case is [tex]\( f(x) \)[/tex] translated 4 units to the right and 6 units up.
#### Option 2: [tex]\( g(x) = (x+6)^2 - 4 \)[/tex]
1. Horizontal Translation: In [tex]\( (x+6)^2 \)[/tex], the [tex]\( x \)[/tex] is replaced by [tex]\( x+6 \)[/tex]. This translates the graph of [tex]\( f(x) = x^2 \)[/tex] horizontally 6 units to the left.
2. Vertical Translation: The [tex]\( -4 \)[/tex] outside the squared term represents a downward translation by 4 units.
So, [tex]\( g(x) \)[/tex] in this case is [tex]\( f(x) \)[/tex] translated 6 units to the left and 4 units down.
#### Option 3: [tex]\( g(x) = (x-6)^2 - 4 \)[/tex]
1. Horizontal Translation: In [tex]\( (x-6)^2 \)[/tex], the [tex]\( x \)[/tex] is replaced by [tex]\( x-6 \)[/tex]. This translates the graph of [tex]\( f(x) = x^2 \)[/tex] horizontally 6 units to the right.
2. Vertical Translation: The [tex]\( -4 \)[/tex] outside the squared term represents a downward translation by 4 units.
So, [tex]\( g(x) \)[/tex] in this case is [tex]\( f(x) \)[/tex] translated 6 units to the right and 4 units down.
#### Option 4: [tex]\( g(x) = (x+4)^2 + 6 \)[/tex]
1. Horizontal Translation: In [tex]\( (x+4)^2 \)[/tex], the [tex]\( x \)[/tex] is replaced by [tex]\( x+4 \)[/tex]. This translates the graph of [tex]\( f(x) = x^2 \)[/tex] horizontally 4 units to the left.
2. Vertical Translation: The [tex]\( +6 \)[/tex] outside the squared term represents an upward translation by 6 units.
So, [tex]\( g(x) \)[/tex] in this case is [tex]\( f(x) \)[/tex] translated 4 units to the left and 6 units up.
### Conclusion
Among the given options:
- The transformation that translates [tex]\( f(x) = x^2 \)[/tex] horizontally 4 units to the right and vertically 6 units up is found in Option 1.
Thus, the equation of the translated function [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = (x-4)^2 + 6 \][/tex]
So, the correct answer is:
[tex]\[ g(x) = (x-4)^2 + 6 \][/tex]
