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Exponentially Decreasing Function — Definition, Formula & Examples

An exponentially decreasing function is a function whose output gets smaller by the same percentage or ratio over each equal interval of the input. As the input increases, the output shrinks rapidly at first and then levels off, approaching zero without ever reaching it.

A function f(x)=a⋅bxf(x) = a \cdot b^x is exponentially decreasing when the initial value a>0a > 0 and the base satisfies 0<b<10 < b < 1. Equivalently, f(x)=a⋅e−kxf(x) = a \cdot e^{-kx} with a>0a > 0 and k>0k > 0 defines the same family of functions.

Key Formula

f(x)=a⋅bx,0<b<1f(x) = a \cdot b^x, \quad 0 < b < 1
Where:
  • aa = Initial value (the output when x = 0); must be positive
  • bb = Base, or decay factor; strictly between 0 and 1
  • xx = Input variable (often represents time)

How It Works

Each time xx increases by 1, the function's value is multiplied by the base bb. Because 0<b<10 < b < 1, each multiplication makes the value smaller. For example, if b=0.5b = 0.5, the output is cut in half with every unit step. The graph curves downward from left to right, always staying above the xx-axis. The horizontal asymptote is y=0y = 0.

Worked Example

Problem: A sample of 800 bacteria decreases by 25% each hour. Write an exponentially decreasing function for the population and find the population after 3 hours.
Identify the initial value and decay factor: The initial population is 800. A 25% decrease each hour means 75% remains, so the decay factor is 0.75.
a=800,b=1−0.25=0.75a = 800, \quad b = 1 - 0.25 = 0.75
Write the function: Substitute into the exponential form.
f(x)=800⋅0.75xf(x) = 800 \cdot 0.75^x
Evaluate at x = 3: Compute the population after 3 hours.
f(3)=800⋅0.753=800⋅0.421875=337.5f(3) = 800 \cdot 0.75^3 = 800 \cdot 0.421875 = 337.5
Answer: After 3 hours the population is approximately 338 bacteria.

Why It Matters

Exponentially decreasing functions model real scenarios such as radioactive decay, cooling liquids, and depreciation of car value. Recognizing this function type is essential in Algebra 2 and Precalculus when solving exponential equations and interpreting data that follows a decay pattern.

Common Mistakes

Mistake: Using a base greater than 1 and expecting the function to decrease.
Correction: A base greater than 1 produces exponential growth. For exponential decrease, the base must satisfy 0<b<10 < b < 1. If you are given a percentage decrease of rr, compute b=1−rb = 1 - r.