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Graphing Exponential Functions

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Graphing Exponential Functions

Study Guide

Key Definition

An exponential function is defined as $y = a^x$, where $a$ is a constant, a > 0 and a ≠ 1.

Important Notes

The graph of $y = 2^x$ has the x-axis as an asymptote.
Changing the base $a$ changes the shape of the graph.
Replacing $x$ with $-x$ reflects the graph across the y-axis.
Replacing $y$ with $-y$ reflects it across the x-axis.
Translating $x$ inside the exponent by $+h$ ($y = a^{x + h}$) moves the graph $h$ units to the left; translating the output ($y = a^x + h$) shifts it up/down by $h$ units.

Mathematical Notation

$y = a^x$ defines an exponential function

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

Exponential functions model growth and decay processes, with key properties shown in their graphs.

Remember

The x-axis acts as an asymptote for exponential graphs like $y = 2^x$.

Quick Reference

Reflection across y-axis:$y = a^{-x}$

Translation up:$y = a^x + k$

Understanding Graphing Exponential Functions

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

To graph y = 2^x, start by making a table of values: for x = -2, -1, 0, 1, 2, compute y = 1/4, 1/2, 1, 2, 4. Plot these points on the coordinate plane. Observe that as x → -∞, y approaches 0 (the horizontal asymptote), and as x → ∞, y grows rapidly.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which of the following formulas represents exponential growth with base 2?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you deposit $100 in a savings account where the money doubles each year following $y = 100 \times 2^x$. How much money will you have after 5 years?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider how the graph of $y = 2^{x+3}$ differs from $y = 2^x$.

Challenge Quiz

Single Choice Quiz

Advanced

Which transformation results in $y = 2^{-x}$?

Recap

Watch & Learn

Review key concepts and takeaways