Sure, let's determine the equation of the translated function [tex]\( g(x) \)[/tex] if the original function is [tex]\( f(x) = x^2 \)[/tex].
We are given several options for the translated function [tex]\( g(x) \)[/tex]:
1. [tex]\( g(x) = (x-4)^2 + 6 \)[/tex]
2. [tex]\( g(x) = (x+6)^2 - 4 \)[/tex]
3. [tex]\( g(x) = (x-6)^2 - 4 \)[/tex]
4. [tex]\( g(x) = (x+4)^2 + 6 \)[/tex]
### Step-by-Step Analysis:
1. Understanding Transformations:
- A translation of the form [tex]\( f(x - h) \)[/tex] shifts the graph horizontally by [tex]\( h \)[/tex] units.
- If [tex]\( h \)[/tex] is positive, the shift is to the right.
- If [tex]\( h \)[/tex] is negative, the shift is to the left.
- A translation of the form [tex]\( f(x) + k \)[/tex] shifts the graph vertically by [tex]\( k \)[/tex] units.
- If [tex]\( k \)[/tex] is positive, the shift is upward.
- If [tex]\( k \)[/tex] is negative, the shift is downward.
2. Analyzing Each Option:
- Option 1: [tex]\( g(x) = (x-4)^2 + 6 \)[/tex]
- This represents a shift 4 units to the right ("-4") and 6 units up ("+6").
- Option 2: [tex]\( g(x) = (x+6)^2 - 4 \)[/tex]
- This represents a shift 6 units to the left ("+6") and 4 units down ("-4").
- Option 3: [tex]\( g(x) = (x-6)^2 - 4 \)[/tex]
- This represents a shift 6 units to the right ("-6") and 4 units down ("-4").
- Option 4: [tex]\( g(x) = (x+4)^2 + 6 \)[/tex]
- This represents a shift 4 units to the left ("+4") and 6 units up ("+6").
3. Selecting the Correct Option:
- Based on the transformations analyzed, the option that translates the original function [tex]\( f(x) = x^2 \)[/tex] by shifting 6 units to the right and 4 units down is:
- [tex]\( g(x) = (x-6)^2 - 4 \)[/tex]
Therefore, the equation of the translated function [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = (x-6)^2 - 4 \][/tex]
This corresponds to Option 3.
