The Exponential Function
Simple Explanation
An exponential function, y = a·bˣ, has x in the exponent rather than the base. When b > 1, the curve grows increasingly quickly (exponential growth); when 0 < b < 1, it shrinks increasingly quickly toward zero (exponential decay). The x-axis (y=0) is always a horizontal asymptote.
Why Do We Need It?
Exponential functions model any quantity that changes by a fixed percentage over equal time periods — population growth, compound interest, and radioactive-style decay all share this shape.
See It
A curve that starts very close to the x-axis on the left, crosses the y-axis at 1, and rises increasingly steeply to the right
Formula
General Exponential Function
y = a · bˣ (a ≠ 0, b > 0, b ≠ 1)
An exponential function grows (if b>1) or decays (if 0<b<1) increasingly quickly, and always has a horizontal asymptote at y = 0.
- a
- — the y-intercept when x=0 (since bˣ⁼⁰ = 1)
- b
- — the base — determines growth (b>1) or decay (0<b<1)
When to use it: Whenever you need to recognize or sketch the shape of an exponential growth or decay function.
Worked Example
Evaluate an exponential function
For y = 3·2ˣ, find y when x = 4.
Why Does This Work?
Since bˣ is always positive for b>0, the graph of y=a·bˣ (with a>0) always stays above the x-axis, approaching it (but never reaching it, since bˣ never equals exactly 0) as x becomes very negative — which is exactly why y=0 is a horizontal asymptote.
Real-Life Example
Bacterial population growth
A bacteria population doubles every hour, starting from an initial count.
Population = (initial count) × 2ᵗ is a direct exponential function, with b=2 representing the doubling.
Practice
For y = 5·2ˣ, find y when x = 0.
MediumCommon mistake
Confusing an exponential function (x in the exponent, e.g. y=2ˣ) with a power function (x as the base, e.g. y=x²) — these have completely different shapes and behaviors.
Quick Review
- y = a·bˣ, with b>0, b≠1.
- b>1: growth. 0<b<1: decay. Always has asymptote y=0.
- The y-intercept is always a, since bˣ=1 when x=0.