Exponential Function Context and Data Modeling
Olivia King
10 min read
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Study Guide Overview
This study guide covers exponential functions in the context of data modeling, including exponential growth and decay. It explains how to identify and model exponential patterns, even when a constant needs to be added. The guide also discusses finding exponential functions using two input-output pairs (x1, y1), (x2, y2) and using technology for exponential regression. It introduces the natural base e and its significance in continuous growth/decay. Finally, it emphasizes understanding equivalent forms of exponential functions and their interpretations. Practice questions covering these concepts are included.
#2.5 Exponential Function Context and Data Modeling
Exponential functions are a major topic on the AP exam, often appearing in both multiple-choice and free-response questions. Make sure you understand the core concepts and how to apply them!
Exponential functions model growth patterns where successive output values over equal-length input-value intervals are proportional. ๐
As a reminder, the general form of an exponential function is , where 'a' is the initial value and 'b' is the base, a positive number other than 1. The base, 'b', represents the constant proportion by which the output value is multiplied at each step.
- When b > 1, the function shows exponential growth (increasing at an increasing rate).
- When 0 < b < 1, the function shows exponential decay (decreasing at a decreasing rate).
When the input values are whole numbers, exponential functions model situations of repeated multiplication of a constant to an initial value.
#๐คจ What if we add a constant?
In some cases, a data set may not immediately reveal an exponential growth pattern, even though the relationship between the independent and dependent variables is actually exponential. One reason for this is that the initial value of the dependent variable may not be zero. In such cases, a constant may need to be added to the dependent variable values of a data set to reveal a proportional growth pattern.
For example, let's say we have a data set that represents the population of a certain city over time. The data set may not show an exponential growth pattern because the population has always been greater than zero. ๐ง
If we want to analyze the population growth rate, we need to consider the population growth rate as a proportion of the initial population. To do this, we can add a constant, say k, to the population values, so the new dependent variable would be . Now we can analyze the growth rate of y' over time, which will be proportional.
Another example could be an investment that starts at a certain value and the growth rate is not immediately visible until we add a constant to the value of the investment.
A graph with four different percentages of growth rates. From least to most curved (which demonstrates growth) is as follows: three percent, seven percent, fifteen percent, and fifty-five percent.
Source: Seeking Alpha
Look for scenarios where adding a constant can reveal an exponential relationship. This is a common trick in AP problems!
This idea is important in precalculus because it allows us to analyze and understand the growth patterns of real-world data sets, even when the data doesn't reveal the pattern immediately. This can be done by performing mathematical operations on the data set, such as adding a cons...
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