Which of these is an exponential function? \( y=x^{2}+\frac{1}{2} \quad y=\left(\frac{1}{4}\right)^{x} \)

The function \( y=\left(\frac{1}{4}\right)^{x} \) is an exponential function because it has the form \( y = a^{x} \), where \( a \) is a constant. In this case, \( \frac{1}{4} \) is the base raised to the power of \( x \), which defines it as exponential. On the other hand, \( y=x^{2}+\frac{1}{2} \) is a polynomial function, not exponential. Exponential functions are not just about rapid growth; they can also model decay, like radioactivity! A classic example is the famous equation \( N(t) = N_0 e^{-\lambda t} \), which describes how a radioactive substance decreases over time. Now that’s some real-world drama, isn’t it? If you’re diving deeper into functions, consider spending time with resources on calculus and graphing techniques. Understanding concepts like the natural exponential function \( e \) and its properties can elevate your math game. A dazzling place to start could be classic textbooks or interactive online platforms that offer calculus problems and explanations.