Question

"What is the characteristic of an exponential function? (Please explain why)

A. As the x-value increases, an exponential function increases or decreases very slowly.
B. As the x-value increases, an exponential function increases or decreases at a constant rate.
C. As the x-value increases, an exponential function is always represented by a straight line
D. As the x-value increases, an exponential function increases or decreases very quickly."

Answer 1

An exponential function increases or decreases very quickly as the x-value increases because a change in the x-value leads to a proportional and more substantial change in the y-value.

Step-by-step explanation:

The characteristic of an exponential function is its rate of change. As the x-value increases, the rate at which the exponential function increases or decreases is not linear nor constant but grows more rapidly. This behavior is exemplified in scenarios such as bacterial growth, where each generation of bacteria multiplies, significantly increasing the growth rate as the population size becomes larger.

An exponential function's equation typically looks like y = a * b^x, where a is a constant, b is the base of the exponential (b > 0, b ≠ 1), and x is the exponent. When x increases, the quantity b^x increases very quickly if b is greater than 1, leading to a sharp upturn in the value of y. Conversely, if b is between 0 and 1, the function decreases rapidly as x increases.