Exponential Functions

An exponential function places the variable in the exponent rather than the base. This simple shift produces functions with remarkable properties: they grow (or decay) at a rate proportional to their current value, they are defined for all real , and they are always positive. Understanding their graphs and transformations is the foundation for logarithms, compound interest, population models, and calculus.

Quick Reference

Property	Formula / Value
General form	, where ,
Domain
Range
-intercept	for every base
Asymptote	-axis ()
	Increasing (exponential growth)
	Decreasing (exponential decay)
Reflection in -axis
Natural base

Definition and Basic Properties

A function of the form

where is a fixed constant and is the variable is called an exponential function with base . The variable appears in the exponent, which is what makes it "exponential."

The requirement is essential: when , the expression is only defined for specific rational values of (those with odd-integer denominators), so the function would not have a continuous domain. Requiring ensures is defined for all real . The case gives the constant function , which is uninteresting.

Three key properties hold for every exponential function (with , ):

The expression is defined for all real , so the domain of is .
Since for any , the graph of always lies above the -axis; the range is .
Since , the graph passes through : every exponential function shares the same -intercept.

If with , :

Graphs of Exponential Functions

The shape of the graph depends critically on whether or .

When : Exponential Growth

When , the function increases as increases. For large positive , the value of becomes very large. For large negative , the value of approaches zero from above: the graph hugs the negative -axis but never touches it. The -axis is a horizontal asymptote.

Graphs of exponential functions y = bË£ for bases b > 1 (2Ë£, 3Ë£, 1.5Ë£), showing exponential growth. — Graphs of exponential functions when . All pass through and grow without bound as .

When : Exponential Decay

When , the function decreases as increases. As , the value approaches zero; as , the value grows without bound. The range is still .

Graphs of exponential functions y = bË£ for bases 0 < b < 1 (0.5Ë£, 0.2Ë£), showing exponential decay. — Graphs of exponential functions when . All pass through and decay toward zero as .

Symmetry Between Reciprocal Bases

The graphs of and are reflections of each other in the -axis. To see why, note that

The function takes the same values for positive as takes for negative of the same magnitude, and vice versa. This is exactly what reflection in the -axis does.

Graphs showing the symmetry between exponential functions with reciprocal bases (e.g., 2Ë£ and (Â½)Ë£) across the y-axis. — The graphs of and are symmetric with respect to the -axis.

Because whenever , and whenever , every decaying exponential corresponds to a growing one obtained by replacing with .

Exponential Growth Vs. Polynomial Growth

Comparison graph of exponential growth (2Ë£) versus polynomial growth (xÂ²), showing the exponential eventually overtaking the polynomial. — The graphs of and intersect at three points, but for , the exponential dominates and the gap widens steeply.

When , the exponential function eventually outgrows any polynomial , no matter how large is. For example, with and : at , the polynomial equals , while , roughly times larger.

Graphing Transformations of Exponential Functions

Sketch the graph of each function and determine its domain and range.

Solution

(a) Since the base 3 is positive, there is no restriction on

. The domain of

. To graph

, we use transformations starting from

: Step 1. Begin with

Graph of the exponential function y = 3Ë£. — Graph of

Step 2. Reflect in the

-axis to obtain

Graph showing the reflection of y = 3Ë£ across the x-axis to form y = -3Ë£. — Graphs of and

Step 3. Shift upward 1 unit to obtain

Graph showing the vertical shift of y = -3Ë£ up 1 unit to form y = 1 - 3Ë£. — Graphs of and

The range of

. We verify algebraically: since

, we have

(b) Again the domain is

. To graph

: Step 1. Begin with

(same starting graph). Step 2. Reflect in the

-axis to obtain

Graph showing the reflection of y = 3Ë£ across the y-axis to form y = 3â»Ë£. — Graphs of and

Step 3. Shift downward 2 units to obtain

Graph showing the vertical shift of y = 3â»Ë£ down 2 units to form y = 3â»Ë£ - 2. — Graphs of and

The range of

. Verification: since

, a reflection in the

-axis does not change the range, so

, and shifting down 2 gives

The Natural Exponential Function

The natural exponential function is , where

is an irrational constant called Euler's number (also known as Napier's constant). The function is also written .

The number is irrational (like ), and its graph sits between and because :

Graph of the natural exponential function y = eË£ plotted between y = 2Ë£ and y = 3Ë£. — The natural exponential function , with , lies between and .

The natural exponential is called "the" exponential function because it is the unique function satisfying : its rate of change equals its current value at every point. Many calculators have a dedicated key for , and note that .

Frequently Asked Questions

Why can't the base of an exponential function be negative?

When the base

is negative,

is only defined when

is a rational number

with

odd. This leaves the function undefined for irrational

and many rationals, so it cannot be graphed as a continuous curve. Calculus requires functions to be defined for all real

, which is why exponential functions require

What is the horizontal asymptote of an exponential function?

The horizontal asymptote of

is the

-axis, i.e.,

. When

the curve approaches

; when

it approaches

. Transformations shift this asymptote: for example,

has horizontal asymptote

How do I graph transformations of exponential functions?

Use the standard transformation toolkit applied to

: horizontal shift right by
: vertical shift up by
: reflection in the -axis (range becomes )
: reflection in the -axis (equivalent to )
: vertical stretch by factor

Work through transformations step by step, tracking the

-intercept and asymptote at each stage.

Why is the range always

and never negative?

Because

for

, and

for every real

. No matter how large or small

is, the output is always strictly positive. The graph lives entirely above the

-axis. Transformations like

shift or flip the output, but the untransformed base function always produces positive values.

Does the exponential function eventually beat every polynomial?

Yes. For any base

and any polynomial

(no matter how large

is), the exponential

eventually dominates: as

. This is often stated as "exponentials grow faster than any polynomial." The crossover point may be very far to the right for small

or large

, but the exponential always wins in the long run.

What is the significance of

compared to other bases?

The number

is the unique base for which

exactly, with no extra constant factor. For any other base,

, which is only equal to

when

, i.e., when

. This self-referential derivative property makes

the natural choice for modeling continuous growth, solving differential equations, and simplifying integrals throughout calculus.