Final answer:
The exponential function f(x) = 4(0.5)^x + 2 represents exponential decay, where the amount decreases over time. To graph this, the decay rate and mean should be highlighted, and the probability of specific outcomes can be shown by shading the relevant area under the curve.
Step-by-step explanation:
Understanding the Exponential Function
The exponential function described, f(x) = 4(0.5)^x + 2, represents an exponential decay since the base of the exponent, 0.5, is between 0 and 1. When graphing, one would label the x-axis as the independent variable and the y-axis as the dependent variable, f(x). The curve would be downward-sloping, showing decay as x increases. To highlight the probability of a student having less than $0.40 in their pocket, one would shade the area under the curve from x=0 to x=0.40. The decay rate in this function is the exponent base (0.5), which should be labeled on the graph.
For the exponential distribution, the mean or expected value can be defined as the inverse of the rate parameter. Therefore, with a decay rate of 0.5, the mean of this distribution is 2 (since 1/0.5 = 2). This mean value can also be labeled on the graph for clarity. In the context of probability and statistics, an exponential distribution is used to model time until an event occurs, such as the amount of time before the next customer arrives.