01 Key Concepts
General Form
f(x) = a * b^x, where a is the initial value, b is the base (growth/decay factor), and x is the exponent.
Growth vs. Decay
If b > 1, the function grows over time (exponential growth). If 0 < b < 1, it shrinks over time (exponential decay).
Evaluating Exponential Functions
Substitute the given x-value into the exponent, then simplify using the rules of exponents.
The Number e
A special constant (approximately 2.718) that appears naturally in continuous growth and decay models, such as f(x) = a*e^(kx).
Horizontal 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.
02 Key Formulas
- f(x) = a * b^x
- Continuous growth/decay: f(x) = a * e^(kx)
03 Solved Examples
Example 1 A population starts at 100 and doubles every year: P(t) = 100 * 2^t. Find the population after 3 years.
- Substitute t=3: P(3) = 100 * 2^3.
- 2^3 = 8.
- 100 * 8.
Answer: 800
Example 2 Is f(x) = 50 * (0.8)^x growth or decay?
- The base is 0.8, which is between 0 and 1.
Answer: Decay, since 0 < 0.8 < 1
Example 3 A car worth $20,000 depreciates 15% per year: V(t) = 20000*(0.85)^t. Find its value after 2 years.
- Substitute t=2: V(2) = 20000*(0.85)^2.
- (0.85)^2 = 0.7225.
- 20000 * 0.7225.
Answer: $14,450
04 Practice Questions
1For f(x)=2*3^x, find f(2).
18
2Is f(x)=10*(1.05)^x growth or decay?
Growth, since 1.05 > 1
3Is f(x)=100*(0.5)^x growth or decay?
Decay, since 0 < 0.5 < 1
4For f(x)=5*2^x, find f(0).
5
5What value does a*b^x (with 0<b<1) approach as x gets very large?
0 (approaches the horizontal asymptote y=0)
📄 Exponential Functions — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.