Exponential growth

Definition

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.

5 Must Know Facts For Your Next Test

1. 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.
2. In exponential growth, if $b > 1$, then the function will increase as $x$ increases.
3. The graph of an exponential growth function is a J-shaped curve that becomes steeper as it moves to the right.
4. Exponential growth functions have horizontal asymptotes along the x-axis (as $y$ approaches zero).
5. 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 $y = a \cdot b^x$.

Horizontal Asymptote: A horizontal line that a graph approaches but never touches or crosses as $x$ tends to infinity or negative infinity.