Exponential Growth and Decay
Many real-world scenarios can be modeled by exponential functions. Exponential growth is an increase in a quantity by a constant percent rate per unit of time. Scenarios that exhibit exponential growth, such as bacteria growth, interest accrual, and population growth, are modeled by increasing exponential functions.
An exponential growth function, where $a>0$, has the form:Determine the number of members expected in 2023.
The number of years between 2018 and 2023 is 5, so $t=5$.Compound interest is a common application of exponential growth. When an account accrues compound interest, the interest is calculated on both the principal (the initial amount of money invested or lent) and on any previously earned interest.
The compound interest function has the form:The annual rate of growth is 3.5%, so $r=0.035$.
Compounded monthly means 12 times per year, so $n=12$.Determine the value of the account after eight years.
The time in years is eight, so $t=8$.Exponential decay is a decrease in a quantity by a constant percent rate per unit of time. Scenarios that exhibit exponential decay, such as radioactive decay and depreciation, are modeled by decreasing exponential functions.
An exponential decay function, where $a>0$, has the form:Exponential Modeling
A trend observed in a set of real-world data may not be linear. An exponential relationship is a nonlinear relationship for which an exponential function is an appropriate model. If the pattern formed by a set of data points approximates an exponential growth curve or an exponential decay curve, an exponential model is appropriate.
Exponential regression is the mathematical process for finding the equation of an exponential function that best models a data set. Exponential regression is often performed using technology. A model generated by exponential regression can be used to make predictions.
The data set in the table represents the number of bacteria in a scientist's sample over time. Use technology to determine an exponential model for the data set. Then use the model to identify the expected number of bacteria after eight hours.
Time in hours ($x$) | 1 | 2 | 3 | 4 | 6 |
Number of Bacteria ($f(x)$) | 346 | 1,218 | 2,254 | 4,179 | 14,850 |
Enter the data into a graphing calculator, spreadsheet, or computer program, and identify the equation of a function to model the data by using exponential regression.
The utility may give values for $a$ and $b$, as well as values for $r^2$ and $r$. Use the values of $a$ and $b$ to write the function that models the data. The variable $r$ represents the correlation coefficient, which indicates the strength of the correlation. The value of $r$ is not used in the exponential function.