Exponential Functions
Graph and solve exponential growth and decay problems
Start with the basics and progress through 10 lessons. Each lesson builds on the previous one.
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In This Topic (10 lessons)
Introduction to Exponential Functions
Learn what exponential functions are, how they differ from linear and polynomial functions, and why they model explosive growth and decay.
Exponential Growth
Learn how quantities grow exponentially and apply growth formulas to real-world scenarios.
Graphing Exponential Functions
Plot y = a·b^x from a table of values, read its intercept and asymptote, and see how shifts, reflections and stretches move the graph.
Exponential Decay
Learn how quantities shrink by a constant percentage, use the decay formula y = a(1 - r)^t and solve depreciation, medicine and cooling problems.
The Natural Base e
Discover where the number e comes from, why it is about 2.71828, and how the natural exponential function y = e^x behaves.
Half-Life Problems
Learn how to use half-life to find how much of a substance remains, how much time has passed, and how long a half-life is.
Continuous Growth and Decay
Use A = Pe^(rt) and A = Pe^(-kt) to model continuous compounding, bacteria growth and radioactive decay, and compare continuous with annual compounding.
Solving Exponential Equations
Solve equations with the variable in the exponent by writing both sides as powers of the same base, and estimate solutions that are not whole numbers.
Population Growth Models
Model growing populations of bacteria, cities and animals with exponential functions, doubling time and the rule of 70.
Exponential Function Word Problems
Solve real exponential word problems with one 5-step method: compound interest, depreciation, viral spread, half-life and continuous growth.
Exponential functions model rapid growth or decay where the rate of change is proportional to the current amount. From compound interest to population growth to radioactive decay, these functions describe many natural phenomena.
What You'll Learn
- Graph exponential growth and decay functions
- Identify key features of exponential graphs
- Write exponential equations from context
- Solve exponential equations
- Apply to real-world growth and decay problems
Frequently Asked Questions
What makes a function exponential?
In exponential functions, the variable is in the exponent: $f(x) = a \cdot b^x$. The base b determines growth (b>1) or decay (0<b<1).
What is the difference between linear and exponential growth?
Linear growth adds a constant amount each period. Exponential growth multiplies by a constant factor, causing much faster increases.
What is the natural exponential function?
$f(x) = e^x$ where e ≈ 2.718. It appears naturally in continuous growth and calculus.