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 a mathematical expression of the form y = a^x, where a is a constant and x is the variable. In contrast, a polynomial function is a mathematical expression of the form y = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where a_n, a_(n-1), ..., a_1, a_0 are constants and n is a non-negative integer.
One key difference between exponential and polynomial functions is the number of possible zeros. An exponential function can only have one zero, which is when the base a is equal to 1. For example, y = 2^x has a single zero at x = 0. In contrast, a polynomial function can have multiple zeros depending on its degree. For example, the polynomial function y = x^2 - 4x + 4 has two zeros at x = 2.
In terms of graph behavior, exponential functions typically start at the y-axis and grow or decay rapidly depending on the base value. The graphs of exponential functions are also asymptotic to the x-axis. In contrast, polynomial functions can have a variety of graph shapes depending on their degree. For example, a quadratic function y = x^2 - 4x + 4 forms a parabola with a vertex at (2, 0).
Real-world examples of exponential functions include population growth, radioactive decay, and compound interest. For instance, the growth of bacteria in a petri dish can be modeled by the exponential function y = 2^x, where x is the time in hours and y is the population count. Real-world examples of polynomial functions include modeling the trajectory of a projectile, the shape of a bridge, and the curve of a roller coaster. For instance, the height of a projectile can be modeled by the quadratic function y = -16x^2 + 20x, where x is the time in seconds and y is the height in feet.