To determine the correct equation representing the transformation of the parent quartic function [tex]\( f(x) = x^4 \)[/tex], let's go through the translation step-by-step.
1. Translation 3 units to the right:
- When a function is translated horizontally to the right by [tex]\( h \)[/tex] units, the transformation involves shifting the variable [tex]\( x \)[/tex] to [tex]\( x-h \)[/tex]. In this case, [tex]\( h = 3 \)[/tex].
- Therefore, the function [tex]\( f(x) = x^4 \)[/tex] becomes [tex]\( f(x-3) = (x-3)^4 \)[/tex].
2. Translation 1 unit down:
- When a function is translated vertically downward by [tex]\( k \)[/tex] units, the transformation involves subtracting [tex]\( k \)[/tex] from the function [tex]\( f(x) \)[/tex]. In this case, [tex]\( k = 1 \)[/tex].
- Therefore, the function [tex]\( (x-3)^4 \)[/tex] translates downward to become [tex]\( (x-3)^4 - 1 \)[/tex].
By combining these two transformations, we get the new function:
[tex]\[ g(x) = (x-3)^4 - 1 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{g(x) = (x-3)^4 - 1} \][/tex]
This corresponds to option D.
Thus, the equation that represents the transformation of translating the [tex]\( y \)[/tex]-intercept of the parent quartic function [tex]\( f(x) = x^4 \)[/tex] 3 units to the right and 1 unit down is option D:
[tex]\[ \boxed{g(x) = (x-3)^4 - 1} \][/tex]
