Problem 38 - Algebra 2

Question

For each function, find the percent increase or decrease that the function models. $$ f(x)=2(0.65)^{x} $$

Step-by-Step Solution

Answer

The function models a 35% decrease.

Step 1: Identify the Percent Change

The function given is of the form f(x) = a(b)^x, where a represents the initial amount and b represents the base of the exponential function. In this function, the base b is 0.65, which means the function represents a decrease. To find the percent decrease, subtract b from 1 and multiply the result by 100.

Step 2: Calculate the Percent Change

Subtract 0.65 from 1 to find the change, and then multiply by 100 to convert this change into a percentage. So, the percent change is (1 - 0.65) \(\times\) 100.

Step 3: Calculate and Interpret the Result

Perform the calculation: (1 - 0.65) \(\times\) 100 = 0.35 \(\times\) 100 = 35%. This means that the function models a 35% decrease per unit increase in x.

Key Concepts

Exponential Decay

When we discuss exponential decay, we're looking at a process where something decreases by a consistent percentage over equal intervals of time or space. This is perfectly modeled by the function f(x)=2(0.65)^{x} from the exercise. In real-life scenarios, exponential decay can be observed in radioactive decay, where radioisotopes lose mass, and in depreciation of the value of cars or electronics over time.

The amount of decay is dictated by the base of the exponential function, in this case, 0.65. Since this number is less than 1, it's a clear sign of decay. Each time we move one unit forward in x, the function's value multiplies by the base (0.65 here), leading to a smaller and smaller result, which is the essence of exponential decay.

Percent Decrease

Understanding percent decrease is vital when interpreting functions that exhibit exponential decay. A percent decrease measures how much a quantity has reduced in proportion to its original amount. Typically, it's expressed using a percentage.

In our function f(x)=2(0.65)^{x}, the percent decrease per unit increase in 'x' is found by the formula \( (1 - \text{base}) \times 100% \). Here, the base is 0.65, and the calculation \( (1 - 0.65) \times 100% \) yields a 35% decrease per unit increase in x. This percentage gives us a standardized way of understanding how quickly the function's value is going down, regardless of the function's specific context or units of measure.

Base of Exponential Function

The base of an exponential function determines the function's growth or decay rate. For the exponential function written as \(f(x) = a(b)^{x}\), 'a' is the initial amount and 'b' is the base. When 'b' is greater than 1, the function represents growth; when 'b' is less than 1, as it is in our function \(f(x)=2(0.65)^{x}\), we're dealing with decay.

The base tells us the factor by which the quantity changes for each increase in 'x'. For example, a base of 0.65 means the quantity is multiplied by 0.65 for every step increase in 'x', leading to a smaller result. Hence, the base plays a crucial role in the function's behavior over its domain and is key to understanding how exponential functions are modeled in various fields such as finance, population studies, and natural sciences.