(a) \( y=\frac{4 x^{5}}{5} \)

Did you know that polynomial functions like \( y=\frac{4 x^{5}}{5} \) have been studied since ancient times? Ancient mathematicians were already exploring concepts of curves and areas, leading to the development of calculus centuries later. This specific function is a fifth-degree polynomial, which means it can have up to five real roots – a thrilling exploration for graph enthusiasts! In the real world, polynomial functions are often used in physics and engineering to model various phenomena, such as motion, where position can relate to time in polynomial terms. This particular function could represent the relationship in a system where forces act on an object, illustrating how its position changes drastically with respect to a variable, like speed or time, under certain conditions. Engaging, isn’t it?