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. short answer
An exponential function is a function of the form f(x) = a^x, where a is a constant greater than zero and not equal to 1. 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_0 are constants and n is a non-negative integer.
1. Zeros:
Exponential functions can only have one zero, which occurs when the base a is greater than 1 and the exponent x is negative. For example, the exponential function f(x) = 2^x has a zero at x = 0 because 2^0 = 1, but it doesn't have any other zeros. Polynomial functions, on the other hand, can have multiple zeros depending on their degree. For instance, the polynomial function f(x) = x^2 - 4x + 4 has a zero at x = 2, since when x = 2, the equation becomes 2^2 - 4(2) + 4 = 4 - 8 + 4 = 0. Moreover, it has another zero at x = 2 because it's a quadratic function.
2. Graph behavior:
Exponential functions have a distinct graph behavior, where they either increase or decrease exponentially and have a horizontal asymptote. When a > 1, the exponential graph increases as x increases, and when 0 < a < 1, the graph decreases as x increases. On the other hand, polynomial functions have more flexibility in their graph behavior. Depending on their degree and coefficients, polynomial functions can have ups and downs, positive or negative slopes, and varying concavity. For example, the exponential function f(x) = 2^x increases rapidly as x increases, while the polynomial function f(x) = x^2 - 4x + 4 is a downward-opening parabola.
3. Real-world examples:
Exponential functions commonly occur in scenarios involving exponential growth or decay. For instance, population growth, compound interest calculations, or the spread of diseases can be modeled using exponential functions. On the other hand, polynomial functions are often used to represent various real-world phenomena, such as distance-time relationships in physics, economic models, or the shape of objects. An example would be a polynomial function to represent the distance traveled by a car over time, taking into account factors like acceleration, speed limits, and road conditions.