Exponential growth

Definition

Exponential growth describes a process where the quantity increases at a rate proportional to its current value. This results in the quantity growing faster and faster over time.

5 Must Know Facts For Your Next Test

1. The general form of an exponential growth function is $f(x) = a \cdot b^x$, where $a > 0$ and $b > 1$.
2. In an exponential growth model, as $x$ increases, $f(x)$ approaches infinity if $b > 1$.
3. The base $b$ in an exponential function determines the rate of growth; larger values of $b$ result in faster growth.
4. Exponential growth functions have horizontal asymptotes at $y = 0$.
5. The graph of an exponential growth function is always increasing and has a J-shaped curve.

Review Questions

• What is the general form of an exponential growth function?
• How does the base $b$ affect the rate of growth in an exponential function?
• What is the horizontal asymptote of an exponential growth function?

Related terms

Logarithmic Function: A logarithmic function is the inverse of an exponential function and can be written as $y = \log_b(x)$, where $b > 0$ and $b \neq 1$.

Decay: Exponential decay describes a process where the quantity decreases at a rate proportional to its current value, resulting in a rapid decrease initially which slows over time.

$e$ (Euler's Number): $e \approx 2.71828$, it is an irrational number that serves as the base for natural logarithms and appears frequently in continuous compound interest problems.