Lesson 7.1

Graphs of Exponential Functions

Exponential functions describe processes where the rate of change is proportional to the current state—a fundamental principle governing everything from bacterial growth to compound interest.

Introduction

Until now, we have mostly dealt with algebraic functions where the variable is the base, such as or . These are called power functions. In this lesson, we flip the script: the base is a constant, and the variable is in the exponent.

Prerequisite Connection

You should be comfortable with exponent rules, especially negative exponents: . This is crucial for understanding decay graphs.

Today's Increment

We define the exponential function family and analyze its unique behaviors, particularly its rapid growth and asymptotic approach to zero.

Why This Matters

In calculus, exponential functions are unique because their rate of change is proportional to their current value. This property makes them the key to solving differential equations that model everything from population dynamics to heat transfer.

Key Concepts

Definition

An exponential function is a function of the form , where is a positive real number other than 1 ().

The base determines whether the function grows or decays.

The domain is all real numbers .

The range is strictly positive numbers (unless shifted vertically).

Exponential Growth ()

When the base is greater than 1, the function increases rapidly as increases.

Exponential Decay ()

When the base is a fraction between 0 and 1, the function decreases towards 0.

The Horizontal Asymptote

A defining feature of the basic exponential function is its horizontal asymptote at (the x-axis). No matter how large of a negative number you plug in (for growth) or positive number (for decay), the value will get incredibly close to zero but never actually reach it.

Worked Examples

Example 1: Basic Exponential Graph

Basic

Graph the function . State the domain, range, and asymptote.

Create a table of values

Include negative values, zero, and positive values.

x	f(x) = 2^x	Point
-2		(-2, 0.25)
-1		(-1, 0.5)
0		(0, 1)
1		(1, 2)
2		(2, 4)

Identify Key Features

Domain: All real numbers . Range: . Horizontal Asymptote: .

Observation

The graph passes through (0, 1) because any non-zero number to the power of 0 is 1.

Example 2: Graphs of Exponential Decay

Intermediate

Graph . Describe how it relates to .

Evaluate key points

At , .
At , .
At , .

Compare with

Since , the graph of is simply the reflection of across the y-axis.

Answer

The graph starts high on the left and approaches on the right. It is a reflection of over the y-axis.

Example 3: Comparing Bases

Advanced

Compare the graphs of and . Which one grows faster? How do they compare for ?

Analyze Positive X

For , a larger base yields a larger result. while . The graph of is much steeper.

Analyze Negative X

For , the larger base produces a smaller fraction. while . Thus is closer to the x-axis on the left side.

Conclusion

shoots up faster on the right but hugs the asymptote tighter on the left. They intersect at .

Common Pitfalls

Decay vs. Negative Base

Do not confuse a decreasing exponential like (which equals ) with a negative base like . Real exponential functions must have a positive base.

The "Zero" Misconception

Students often think can equal 0. It is important to remember that for all real . The graph gets infinitely close to the x-axis but never touches it.

Real-World Application

Viral Growth

Information spreading on social media often follows an exponential model in the early stages. If every person shares a post with 2 friends, the number of people who see it grows as (where x is the number of sharing waves).

This is why "going viral" happens so suddenly. The jump from 1,000 to 2,000 takes the same amount of time/steps as the jump from 1 million to 2 million.