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 and a polynomial function are different in several aspects. Let's explore those differences using specific examples.
1) Number of Possible Zeros:
- Polynomial Function: A polynomial function of degree "n" can have at most "n" zeros. For example, the polynomial function f(x) = x^2 - 4 has two zeros: x = 2 and x = -2. The degree of this polynomial is 2, so it can have at most 2 zeros.
- Exponential Function: An exponential function does not have zeros. It means that there is no value of "x" for which the function equals zero. For instance, the exponential function g(x) = 2^x never equals zero, regardless of the value of "x". It continuously grows as "x" increases.
2) Graph Behavior:
- Polynomial Function: The graph of a polynomial function may have various behaviors. For example, the polynomial function f(x) = (x - 1)(x - 2)(x - 3) has a degree of 3, and its graph will exhibit "wiggle" behavior around its zeros, as the curve changes direction at each zero.
- Exponential Function: The graph of an exponential function has a distinct behavior characterized by exponential growth or exponential decay. For instance, the function g(x) = 2^x represents exponential growth. As "x" increases, the value of the function grows rapidly, resulting in a steep upward curve. Conversely, if the base were between 0 and 1 (for example, g(x) = (1/2)^x), the graph would exhibit exponential decay as "x" increases, becoming increasingly close to the x-axis.
3) Real-World Examples:
- Polynomial Function: Polynomial functions are versatile and can represent various real-world scenarios. Consider the polynomial function f(x) = x^3 - 4x^2 + x + 6, which could represent the volume of a box with side length "x" and a known height. The polynomial provides different values (positive, zero, or negative) for different values of "x" related to the volume.
- Exponential Function: Exponential functions often describe scenarios involving growth or decay. For example, the function g(t) = 1000 * 1.1^t could represent the exponential growth of a population, where "t" represents time in years. The exponential factor 1.1 suggests that the population increases by 10% each year. As time passes, the population grows significantly due to the compounding nature of exponential functions.
In summary, exponential functions and polynomial functions differ in terms of the number of possible zeros, graph behavior, and real-world applications. Polynomial functions may have multiple zeros, exhibit various behaviors, and can represent quantities related to volume, while exponential functions have no zeros, show exponential growth or decay, and are commonly used to model population growth, financial investments, or bacterial growth, among other scenarios.