Exponential Growth & Decay

You’ll be able to

Identify the initial value and base of an exponential function
Classify a function as growth or decay from its base
Build an exponential model from a doubling or halving description

The shape of an exponential function

An exponential function has the form f(x) = a·bˣ, where a is the initial value (the output when x = 0) and b is the base (the constant multiplier). Unlike a linear function, which adds a fixed amount each step, an exponential multiplies by b each step. That constant ratio is what makes exponentials climb — or fall — so dramatically.

Growth versus decay

The base b decides the direction. If b > 1, each step multiplies by more than one, so the outputs grow: this is exponential growth. If 0 < b < 1, each step multiplies by a fraction, shrinking the output toward zero: exponential decay. In f(x) = 5(0.8)ˣ, the base 0.8 is between 0 and 1, so the function decays, starting from an initial value of 5.

Exponential model

f(x) = a · bˣ

a = value at x = 0 (initial amount). b > 1 grows, 0 < b < 1 decays. The percent change per step is (b − 1)·100%.

Worked example

For f(x) = 5(0.8)ˣ, state the initial value, whether it grows or decays, and the value at x = 1.

1.Read off the initial value a = 5 — that is f(0), since b⁰ = 1.
2.The base is b = 0.8, which lies between 0 and 1, so the function is exponential decay.
3.Evaluate at x = 1: f(1) = 5(0.8)¹ = 4, a 20% drop from 5.

Answer: Initial value 5, decay (because 0 < 0.8 < 1), and f(1) = 4. Each step multiplies the previous output by 0.8, a 20% decrease.

Checkpoint

In the function f(x) = 5(0.8)ˣ, does the function represent growth or decay, and what is the initial value?

Tip

Convert the base to a percent change: b = 1.05 means +5% per step (growth); b = 0.8 means −20% per step (decay). The rule is percent change = (b − 1)·100%.

Checkpoint

A population doubles every 3 years. If it starts at 200, which model gives the population after t years?

On the exam

For growth described by a doubling/halving time T, use base 2 (or 1/2) with exponent t/T. For a percent rate r per period, use base (1 + r). Matching the base and the exponent to the description is the graded step.

Answer the 2 checkpoints as you read.