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 function of the form f(x) = a^x, where a is a constant. On the other hand, a polynomial function is a function of the form f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0, where a_n, a_{n-1}, ..., a_1, and a_0 are constants.
1. Zeros:
Exponential function: An exponential function does not have any real zeros. The graph of an exponential function only intersects the x-axis at x = 0.
Example: f(x) = 2^x. The graph of f(x) = 2^x never crosses the x-axis.
Polynomial function: A polynomial function can have zero, one, or multiple real zeros. The number of zeros is at most equal to the degree of the polynomial.
Example: f(x) = x^2 - 9. This polynomial function has two real zeros, x = -3 and x = 3.
2. Graph behavior:
Exponential function: The graph of an exponential function can either be increasing or decreasing, depending on the value of a.
Example: f(x) = 3^x. The graph of f(x) = 3^x is an increasing curve.
Polynomial function: The graph of a polynomial function can have multiple turning points and can have both increasing and decreasing parts.
Example: f(x) = x^3 - 4x^2 + 3x + 1. The graph of f(x) = x^3 - 4x^2 + 3x + 1 has a turning point and both increasing and decreasing sections.
3. Real-world examples:
Exponential function: Exponential functions are commonly used in finance and biology. In finance, compound interest calculations or population growth models often involve exponential functions.
Example: The growth of a population over time can be modeled using the function f(x) = 2^x, where x represents time and f(x) represents the size of the population at time x.
Polynomial function: Polynomial functions are frequently used to model physical phenomena and data. They can describe relationships like distance traveled, force applied, or temperature changes.
Example: The trajectory of a projectile can be approximated using a polynomial function. The path of a thrown baseball can be modeled using a quadratic function, f(x) = -0.5x^2 + 2x + 1, where f(x) represents the height of the ball at time x.