Master

Exponential Functions

Master exponential functions with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

Exponential Functions

Study Guide

Key Definition

An exponential function is defined as $y = a^x$ where $a$ is a real positive constant (a ≠ 1) and $x$ is a real variable.

Important Notes

If $a > 1$, the function represents growth.
If $0 < a < 1$, the function represents decay.
The domain of $y = a^x$ is all real numbers.
The range of $y = a^x$ is $y > 0$.
The graph of an exponential function approaches, but never intersects, the x-axis (horizontal asymptote y = 0).

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ represents multiplication

$\div$ represents division

$^$ denotes exponentiation

Remember to use proper notation when solving problems

Why It Works

Exponential functions model real-world phenomena where growth or decay occurs at a constant rate, such as population growth or radioactive decay.

Remember

The base $a$ determines the nature of the exponential function: growth or decay.

Quick Reference

Growth Function:$y = a^x$ when $a > 1$

Decay Function:$y = a^x$ when $0 < a < 1$

Understanding Exponential Functions

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Exponential functions look like $y = 2^x$ where the curve rises rapidly (see graph below).

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which of the following is an exponential function?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You invest $100 in a bank with an annual interest rate of 5%. How much will it be in 5 years if compounded annually? Use y = a^x.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Determine the time it takes for a population to double from its initial size P₀ if it follows the function $y = 2^x$.

Challenge Quiz

Single Choice Quiz

Advanced

Find the derivative of $y = 3^x$ with respect to $x$.

Recap

Watch & Learn

Review key concepts and takeaways