Algebra · Quick Reference

Exponential Functions Cheat Sheet

Algebra II · Lesson 11/17

In one line: an exponential function has its variable in the exponent rather than the base — modeling explosive growth or decay, from compound interest to population change to radioactive decay.

Key Ideas

1General Form. f(x) = a * b^x, where a is the initial value, b is the base (growth/decay factor), and x is the exponent.

2Growth vs. Decay. If b > 1, the function grows over time (exponential growth). If 0 < b < 1, it shrinks over time (exponential decay).

3Evaluating Exponential Functions. Substitute the given x-value into the exponent, then simplify using the rules of exponents.

4The Number e. A special constant (approximately 2.718) that appears naturally in continuous growth and decay models, such as f(x) = a*e^(kx).

5Horizontal Asymptote. Exponential functions of the form a*b^x get closer and closer to y=0 as x goes to negative or positive infinity (depending on b), but never actually touch it.

Worked Examples

A population starts at 100 and doubles every year: P(t) = 100 * 2^t. Find the population after 3 years.

800

Is f(x) = 50 * (0.8)^x growth or decay?

Decay, since 0 < 0.8 < 1

A car worth $20,000 depreciates 15% per year: V(t) = 20000*(0.85)^t. Find its value after 2 years.

$14,450

Formulas

f(x) = a * b^x

Continuous growth/decay: f(x) = a * e^(kx)

Practice Yourself

For f(x)=2*3^x, find f(2).

Is f(x)=10*(1.05)^x growth or decay?

Growth, since 1.05 > 1

Is f(x)=100*(0.5)^x growth or decay?

Decay, since 0 < 0.5 < 1