IN 5 SENTENCES How is an exponential function different from a polynomial function? Use specific examples to illustrate your points. Discuss the number of possible zeros, the behavior of the graphs, and possible real-world examples for each.
An exponential function is characterized by a constant base raised to a variable exponent, while a polynomial function consists of terms with non-negative integer exponents on a variable. For instance, f(x) = 2^x is an exponential function, whereas g(x) = x^2 is a polynomial function. Exponential functions typically have one or no zeros, as shifting the graph along the x-axis would correspond to a scaling of the function. They exhibit rapid growth or decay, depending on whether the base is greater than or less than 1. An example of a real-world exponential function can be seen in the population growth of a species. Polynomial functions, on the other hand, can have multiple zeros depending on their degree. The behavior of their graphs may include turning points, or changes in direction, called extrema. An example of a polynomial function is h(x) = x^3 - 2x^2 + x, which could represent the path of a projectile.
