Easy

The Graph of an Exponential Function

Simple Explanation

y = bˣ (with b>0, b≠1) is an exponential function — it grows (or decays, if 0<b<1) by a constant multiplicative factor for every unit increase in x, always passes through (0,1), is always strictly positive, and has a horizontal asymptote at y=0.

Why Do We Need It?

Exponential growth and decay describe an enormous range of real phenomena — population growth, radioactive decay, compound interest, and more — making this one of the most practically important function families.

See It

The graph of y = 2ˣ

Formula

Properties of Exponential Functions

y = bˣ (b>0, b≠1): domain all reals, range (0,∞), y-intercept (0,1), asymptote y=0

An exponential function grows (or decays, if 0<b<1) by a constant multiplicative factor for every unit increase in x, and is always strictly positive.

b: — the base of the exponential function, a positive number not equal to 1

When to use it: Whenever identifying or sketching the basic exponential graph and its key features.

Worked Example

Evaluate an exponential function

Evaluate y=2ˣ at x=0, x=3, and x=−2.

Why Does This Work?

Multiplying by b for every unit increase in x is exactly what "constant multiplicative growth" means — starting from y(0)=1, y(1)=b, y(2)=b², and so on, so y(x)=bˣ captures this pattern for every real x.

Real-Life Example

Compound interest on a savings account

Money in a savings account grows by the same percentage every year, compounding on top of the previous year's total.

This is exactly exponential growth — the account balance after x years follows an exponential function of x.

Practice

Evaluate y=3ˣ at x=2.

Easy

Common mistake

Assuming an exponential function can output zero or a negative value — bˣ is always strictly positive, for any real x, whenever b>0.

Quick Review

y=bˣ (b>0, b≠1): domain all reals, range (0,∞).
Always passes through (0,1); horizontal asymptote at y=0.
Models constant multiplicative growth or decay.