Glossary
Asymptote
A line that a curve approaches as it heads towards infinity. For a basic exponential function, the x-axis (y=0) is a horizontal asymptote, meaning the graph gets infinitely close to it but never touches or crosses it.
Example:
As a decaying radioactive substance approaches zero, its graph gets closer and closer to the x-axis, which acts as a horizontal asymptote.
Base (b)
In an exponential function f(x) = a * b^x, 'b' is the growth or decay factor. It determines how quickly the function's value changes.
Example:
For a population that doubles every year, the base (b) of the exponential function would be 2.
Domain
The set of all possible input values (x-values) for which a function is defined. For basic exponential functions, the domain is all real numbers.
Example:
When modeling the growth of a plant over time, the domain would typically be all non-negative real numbers, as time cannot be negative.
Doubling Time
The amount of time it takes for a quantity that is growing exponentially to double in size. It can be calculated using the natural logarithm.
Example:
If a bacterial population has a doubling time of 20 minutes, it means the number of bacteria will be twice as large every 20 minutes.
Exponential Decay
A pattern of decrease where the quantity diminishes at a rate proportional to its current size. This occurs in f(x) = a * b^x when the base 'b' is between 0 and 1 (0 < b < 1).
Example:
The decrease in the amount of a radioactive substance over time is a classic example of exponential decay.
Exponential Function
A function where the independent variable (x) appears as an exponent, typically following the form f(x) = a * b^x. It models situations of rapid growth or decay.
Example:
The spread of a new viral trend on social media can often be modeled by an exponential function, showing rapid initial growth.
Exponential Growth
A pattern of increase where the quantity grows at a rate proportional to its current size. This occurs in f(x) = a * b^x when the base 'b' is greater than 1.
Example:
The rapid increase in the number of users on a popular new app often demonstrates exponential growth.
Half-life
The amount of time it takes for a quantity that is decaying exponentially to reduce to half of its initial amount. It is commonly used in radioactive decay.
Example:
Carbon-14 has a half-life of approximately 5,730 years, meaning half of any given amount will decay in that time.
Initial Value (a)
In an exponential function f(x) = a * b^x, 'a' represents the starting quantity or the y-intercept of the graph. It's the value of f(x) when x = 0.
Example:
If a savings account starts with 500 is the initial value (a) in the exponential growth equation.
Natural Logarithm (ln)
A logarithm with base 'e' (Euler's number, approximately 2.718). It is often used in calculations involving continuous growth or decay, and for solving for exponents in exponential equations.
Example:
To find the exact time it takes for an investment to reach a certain value with continuous compounding, you would typically use the natural logarithm.
Percentage to Decimal Conversion
The process of converting a percentage rate into its decimal equivalent by dividing by 100. This is crucial for correctly calculating the base 'b' in exponential functions.
Example:
A 5% growth rate must undergo a percentage to decimal conversion to become 0.05 before being added to 1 to form the base (1.05).
Range
The set of all possible output values (y-values) that a function can produce. For an exponential function f(x) = a * b^x (with a > 0), the range is all positive real numbers.
Example:
The number of bacteria in a culture, modeled exponentially, will always have a positive range, as you can't have a negative number of bacteria.
Unit Conversions
The process of converting a measurement from one unit to another (e.g., days to years, grams to kilograms) to ensure consistency within a problem.
Example:
When calculating compound interest annually, if the interest rate is given monthly, you'll need to perform unit conversions to match the time periods.
Y-intercept
The point where the graph of a function crosses the y-axis. For an exponential function f(x) = a * b^x, the y-intercept is always 'a' (when x=0).
Example:
If a graph of car depreciation starts at 25,000 is its y-intercept, representing the car's initial value.