1. The problem asks to classify the function family for the equation $y = 3^{4x}$. 2. This is an exponential function because the variable $x$ is in the exponent. 3. The general form of an exponential function is $y = a^{bx}$ where $a > 0$ and $a \neq 1$. 4. Here, $a = 3$ and $b = 4$, so it fits the exponential function family. 5. Exponential functions have important properties: they grow or decay rapidly depending on the base $a$. 6. Since $a = 3 > 1$, this function is an exponential growth function. Final answer: The function $y = 3^{4x}$ belongs to the \textbf{exponential function family}.