Sure, let’s work through this step-by-step.
We start with the function [tex]\( f(x) = 4x^2 + 6 \)[/tex].
We are asked to translate this function 4 units down. Translating a function down by a specific number involves subtracting that number from the function.
To translate [tex]\( f(x) \)[/tex] downward by 4 units, we subtract 4 from [tex]\( f(x) \)[/tex]:
[tex]\[ g(x) = f(x) - 4 \][/tex]
Now, we plug in the expression for [tex]\( f(x) \)[/tex]:
[tex]\[ g(x) = (4x^2 + 6) - 4 \][/tex]
Simplify the expression by combining like terms:
[tex]\[ g(x) = 4x^2 + 6 - 4 \][/tex]
[tex]\[ g(x) = 4x^2 + 2 \][/tex]
So, the equation for [tex]\( g(x) \)[/tex] in its simplest form is:
[tex]\[ g(x) = 4x^2 + 2 \][/tex]
Thus, [tex]\( g(x) = \boxed{4x^2 + 2} \)[/tex].
