1. \( \left\{(x, y) \mid y=(x-1)^{5}+4\right\} \)

This equation represents a curve known as a quintic function, which is more complex than parabola or cubic functions. The term \( (x-1)^5 \) means that the curve exhibits a distinct turning point at \( x=1 \). It generally simulates an "S" shape, moving from left to right, with a single inflection point where the curve transitions from concave to convex around \( y=4 \). One real-world application of such functions is in modeling population dynamics or the growth of certain species where certain conditions lead to exponential growth or decline. Understanding the behavior of curves like this helps in predicting future trends, making it a powerful tool for ecologists and economists alike—much like a roller coaster's thrilling ups and downs, shifts in behavior can often lead to surprising outcomes!