Glossary
Base
In an exponential function $f(x) = ab^x$, 'b' is the constant proportion by which the output value is multiplied at each step, indicating the growth or decay factor.
Example:
In a population model , the base of 1.03 means the population grows by 3% each time period.
Best-fit function
The exponential function determined by regression that most accurately represents a given data set by minimizing the differences between the observed data points and the function's predicted values.
Example:
After plotting data for a cooling object, the calculator provides a best-fit function that closely matches the temperature decline.
Constant (added to dependent variable)
A numerical value that may be added to the dependent variable values of a data set to reveal an underlying proportional growth pattern, especially when the initial value is not zero.
Example:
If a temperature sensor always reads 5 degrees higher than the actual temperature, adding a constant of -5 to the readings would reveal the true temperature pattern.
Continuous decay
A type of exponential decay modeled using the natural base *e*, where the rate of decrease is applied constantly and instantaneously over time.
Example:
The radioactive decay of an isotope is often modeled as continuous decay.
Continuous growth
A type of exponential growth modeled using the natural base *e*, where the rate of increase is applied constantly and instantaneously over time.
Example:
The growth of a population without any limiting factors can be approximated by a continuous growth model.
Correlation coefficient (R^2)
A statistical measure, typically denoted as R^2, that indicates how well an exponential model fits the data, with values closer to 1 indicating a stronger fit.
Example:
An R^2 value of 0.98 for an exponential model of bacterial growth suggests a very strong fit to the observed data.
Equivalent forms (of exponential functions)
Different algebraic expressions of the same exponential function that have identical graphs and behavior but may reveal different interpretations of the growth or decay rate based on the chosen time unit.
Example:
The function can be written in an equivalent form to show the annual growth rate from a monthly rate.
Exponential decay
Occurs in an exponential function $f(x) = ab^x$ when the base 'b' is between 0 and 1, causing the function's output to decrease at a decreasing rate.
Example:
The amount of a radioactive substance remaining over time demonstrates exponential decay.
Exponential functions
Functions that model growth patterns where successive output values over equal-length input-value intervals are proportional, generally expressed as $f(x) = ab^x$.
Example:
The number of social media followers for a new influencer might follow an exponential function, doubling every month.
Exponential growth
Occurs in an exponential function $f(x) = ab^x$ when the base 'b' is greater than 1, causing the function's output to increase at an increasing rate.
Example:
A rapidly spreading virus might show exponential growth in the number of infected individuals.
Exponential regression
A statistical method used to find the best-fit exponential function that models a given set of data points, minimizing the difference between observed and predicted values.
Example:
A scientist might use exponential regression to model the relationship between the concentration of a drug and its effect over time.
Initial value
In an exponential function $f(x) = ab^x$, 'a' represents the starting quantity or the value of the function when the input (x) is zero.
Example:
If a savings account starts with 500 is the initial value of the investment.
Input-output pairs
Specific data points (x, y) that can be used to construct an exponential function by setting up and solving a system of equations for 'a' and 'b'.
Example:
Given that a plant is 2 cm tall on day 0 and 6 cm tall on day 1, these two input-output pairs (0,2) and (1,6) can define its exponential growth.
Natural base e
An irrational mathematical constant, approximately 2.718, often used as the base in exponential functions that model continuous growth or decay.
Example:
Financial models for continuously compounded interest frequently use the natural base e.
Natural logarithm (ln)
The inverse function of the exponential function with base *e*, denoted as *ln(x)*, used to solve equations involving *e* and to analyze continuous growth/decay.
Example:
To find the time it takes for an investment to double with continuous compounding, you would use the natural logarithm.
Residuals
The differences between the observed (actual) data points and the corresponding predicted values from a regression model, used to assess the model's fit.
Example:
Plotting the residuals from an exponential regression can help identify if the model systematically over- or under-predicts values.
Technology (for exponential regression)
Software or calculators used to perform non-linear regression, specifically to fit an exponential function to a set of data points.
Example:
Using a graphing calculator's statistical functions to perform technology based exponential regression on population data.