Graphing Exponential Functions

Q: How do I tell growth from decay?

Look at the base $b$. Greater than 1 means growth; between 0 and 1 means decay.

Q: What is the $y$-intercept of $a \cdot b^x$?

It is $a$, because $b^0 = 1$ makes $f(0) = a$.

Q: Why doesn't the graph touch the x-axis?

An exponential approaches the horizontal asymptote $y = 0$ but never reaches it — the output gets arbitrarily small without hitting zero.

An exponential function $f(x) = a \cdot b^x$ either grows or decays depending on the base $b$ . A base above $1$ grows; a base between $0$ and $1$ decays toward zero.

The $y$ -intercept is always $a$ , since $b^0 = 1$ . Those two facts — growth-or-decay from $b$ , $y$ -intercept from $a$ — describe the graph.

Growth versus decay

If $b > 1$ , outputs increase as $x$ increases — exponential growth. If $0 < b < 1$ , outputs shrink toward zero — exponential decay.

For $f(x) = 3\cdot\left(\dfrac{2}{5}\right)^x$ , the base $\dfrac{2}{5}$ is between $0$ and $1$ , so the graph decays.

A base greater than $1$ curves the other way: the growth graph $y = 2^x$ below rises faster and faster to the right while flattening toward the x-axis on the left.

The $y$ -intercept and asymptote

At $x = 0$ , $b^0 = 1$ , so $f(0) = a$ . The $y$ -intercept is the coefficient $a$ — here $3$ .

Exponential graphs hug a horizontal asymptote (usually $y = 0$ ) on one side, getting close but never touching it.

Worked examples

Example 1: decay and $y$ -intercept

Describe $f(x) = 3\cdot\left(\dfrac{2}{5}\right)^x$ and give its $y$ -intercept.

Base between 0 and 1

\text{decay}

y

-intercept is a

f(0) = 3

Answer: Exponential decay; $y$ -intercept $(0, 3)$

Example 2: growth or decay

Does $f(x) = 2 \cdot 3^x$ grow or decay?

Base is greater than 1

b = 3 > 1

So it grows

\text{growth}

Answer: Growth

Try one yourself

Common questions

How do I tell growth from decay?

Look at the base $b$ . Greater than 1 means growth; between 0 and 1 means decay.

What is the

y

-intercept of

a \cdot b^x

It is $a$ , because $b^0 = 1$ makes $f(0) = a$ .

Why doesn't the graph touch the x-axis?

An exponential approaches the horizontal asymptote $y = 0$ but never reaches it — the output gets arbitrarily small without hitting zero.

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Graphing Exponential Functions

Growth versus decay

The $y$ -intercept and asymptote

Worked examples

Example 1: decay and $y$ -intercept

Example 2: growth or decay

Try one yourself

Common questions

Keep learning

Special Exponential Functions (the Number e)

Modeling Data: Choosing a Model & Regression

Introduction to Exponential Functions

Graphing Exponential Functions

Growth versus decay

The yyy-intercept and asymptote

Worked examples

Example 1: decay and yyy-intercept

Example 2: growth or decay

Try one yourself

Common questions

Keep learning

Special Exponential Functions (the Number e)

Modeling Data: Choosing a Model & Regression

Introduction to Exponential Functions

The $y$ -intercept and asymptote

Example 1: decay and $y$ -intercept