Final answer:
To analyze an exponential decay graph, assess the initial value or mean (m), label axes and decay rate on the graph, and shade the area under the curve to represent probabilities for certain ranges. In circuits, the time constant (T = RC) dictates the rate at which voltage decays.
Step-by-step explanation:
To analyze the graph of an exponential decay function, we focus on the initial value, decay rate, and the mean of the function. For example, if we have a function given by f(x) = 0.25e(-0.25x), we can note that when x = 0, f(x) equals the initial value or mean, m, which is 0.25.
The decay rate is the coefficient of x in the exponent. In the example, the decay rate is 0.25. On the graph, this rate determines how quickly the function decreases as x increases. To graph the function, label the x-axis as time or another appropriate variable and the y-axis as the value of the function. Moreover, the maximum value of the y-axis will be the initial value m
For exponential decay in an electrical circuit, the time constant T is crucial and is defined as T = RC, where R is resistance and C is capacitance. During the first time interval T, the voltage across a discharging capacitor will decrease to 0.368 of its initial value V0.
When dealing with probability distributions, the area under the curve can represent a probability. For instance, if X follows an exponential distribution with a decay rate of 0.1, to find P(x < 6), you would shade the area under the graph from x = 0 to x = 6 and calculate the integral to determine the actual probability.