Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value. This results in the function increasing rapidly over time.
congrats on reading the definition of exponential growth. now let's actually learn it.
Exponential growth can be modeled by the equation $y = a \cdot b^x$, where $a$ is the initial value, $b$ is the base, and $x$ is the exponent.
In exponential growth, if $b > 1$, then the function will increase as $x$ increases.
The graph of an exponential growth function is a J-shaped curve that becomes steeper as it moves to the right.
Exponential growth functions have horizontal asymptotes along the x-axis (as $y$ approaches zero).
Common real-world examples of exponential growth include population growth, compound interest, and certain types of biological processes.
Review Questions
What is the general form of an exponential growth equation?
How does the value of base $b$ affect whether an exponential function represents growth or decay?
What shape does the graph of an exponential growth function typically take?
Related terms
Logarithmic Functions: Functions that are the inverse of exponential functions, typically written as $y = \log_b(x)$ where $b$ is the base.
Base (of an Exponential Function): The constant factor that determines how quickly an exponential function grows or decays; represented by $b$ in equations like $y = a \cdot b^x$.