Sure, let's analyze how to transform the given quartic function [tex]\( f(x) = x^4 \)[/tex] by translating it 3 units to the right and 1 unit down.
### Step 1: Translation 3 Units to the Right
To translate the function [tex]\( f(x) = x^4 \)[/tex] 3 units to the right, we replace [tex]\( x \)[/tex] with [tex]\( x - 3 \)[/tex]. This gives us the new function:
[tex]\[ g(x) = (x - 3)^4 \][/tex]
### Step 2: Translation 1 Unit Down
Next, we translate the function [tex]\( g(x) = (x - 3)^4 \)[/tex] 1 unit down. To do this, we subtract 1 from the entire function:
[tex]\[ g(x) = (x - 3)^4 - 1 \][/tex]
Thus, the equation that represents the transformation of the parent quartic function [tex]\( f(x) = x^4 \)[/tex] by translating it 3 units to the right and 1 unit down is:
[tex]\[ g(x) = (x - 3)^4 - 1 \][/tex]
### Conclusion
From the given options:
A. [tex]\( g(x) = (x - 3)^4 - 1 \)[/tex]
B. [tex]\( g(x) = (x - 1)^4 + 3 \)[/tex]
C. [tex]\( g(x) = (x + 3)^4 - 1 \)[/tex]
D. [tex]\( g(x) = (x + 1)^4 + 3 \)[/tex]
Option A, [tex]\( g(x)=(x-3)^4-1 \)[/tex], correctly represents the transformation. Therefore, the correct answer is:
[tex]\[ \boxed{A} \][/tex]
