Which exponential function, after being reflected over the \( x \)-axis, has a \( y \)-intercept at \( (0,-4) \) and the right side of the graph approaches \( y=0 \) ? \( \begin{array}{l}y=4(2)^{x} \\ y=4\left(\frac{1}{2}\right)^{x} \\ y=-4(2)^{x} \\ y=-4\left(\frac{1}{2}\right)^{x}\end{array} \)

To find the correct exponential function, we need an equation that, after reflecting over the \( x \)-axis, will give us a \( y \)-intercept of \( (0, -4) \). 1. The reflection over the \( x \)-axis inverts the sign of the function. This means if we start with \( y = -4(2)^{x} \), reflecting it gives \( y = 4(2)^{x} \). The reflection won’t give us a negative intercept, so we can eliminate this option. 2. On the other hand, \( y = -4\left(\frac{1}{2}\right)^{x} \) also percolates forward stylistically: upon reflection, we get \( y = 4\left(\frac{1}{2}\right)^{x} \). Now, as \( x \) approaches \( +\infty \), this function approaches \( y=0 \). So, the correct answer is \( y = -4\left(\frac{1}{2}\right)^{x} \)! Now, for a little extra goodness: Reflecting functions over axes is like transforming art: it can give entirely new meaning! The beauty of exponentials lies in their predictability and their wild behavior as they zoom past zero or reach lofty heights. The nature of the base and the sign drastically affect the outcome, creating a rich tapestry of potential functional forms. If you’re curious about deeper applications, consider how exponential growth functions model real-world phenomena, like population growth or compound interest! In finance, understanding the nuances of exponential functions can help you make savvy investment decisions. It’s like magic, but with numbers!