Introduction to Exponential Functions

Q: How is exponential different from a quadratic like $y = x^2$?

In $y = x^2$ the variable is the base and the exponent is fixed. In an exponential like $y = 2^x$ the variable is the exponent. Quadratics make a U-shape; exponentials make a curve that races up or hugs the $x$-axis.

Q: What is a horizontal asymptote?

It is a horizontal line, $y = c$, that the graph approaches but never touches. For $y = a \cdot b^x$ the asymptote is $y = 0$ — the outputs get tiny on one side but never reach zero.

Q: Can $b$ be negative?

No. In Algebra 1 the base must be positive (and not equal to $1$). A negative base would flip signs at every step and would not make a smooth curve.

Q: What does $a$ do to the graph?

It sets the starting height. $y = 5 \cdot 2^x$ and $y = 2^x$ have the same doubling behavior, but one crosses the $y$-axis at $5$ and the other at $1$.

An exponential function is a function where the variable lives in the exponent: $y = a \cdot b^x$ . Instead of adding the same amount each step like a linear function, it multiplies by the same amount each step — and repeated multiplication makes values explode upward or shrink toward zero fast.

Two numbers control everything. The number $a$ is the initial value — the $y$ -value when $x = 0$ . The number $b$ is the base — the factor you multiply by every time $x$ goes up by $1$ . Once you can read $a$ and $b$ off an equation, you know exactly what the function does.

The two parts: $a$ and $b$

In $y = a \cdot b^x$ , plug in $x = 0$ : since $b^0 = 1$ , you get $y = a$ . That is why $a$ is called the initial value — it is where the graph crosses the $y$ -axis.

The base $b$ tells you what happens as $x$ increases. Each time $x$ goes up by $1$ , the output gets multiplied by $b$ . In $y = 3 \cdot 2^x$ , the values are $3, 6, 12, 24, \ldots$ — each one is double the last.

Growth or decay: look at $b$

If $b > 1$ , the function shows growth — multiplying by a number bigger than $1$ makes values increase, faster and faster. If $0 < b < 1$ , the function shows decay — multiplying by a fraction shrinks the values toward zero.

Either way, the graph flattens out along the line $y = 0$ . That line is called the horizontal asymptote: the graph gets closer and closer to it but never touches it. A basic exponential function never outputs zero, because no amount of multiplying by $b$ turns a nonzero number into $0$ .

Exponential vs. linear

A linear function adds the same amount each step — a constant difference. An exponential function multiplies by the same factor each step — a constant ratio. On a graph, linear is a straight line; exponential is a curve that starts slow and then takes off (or starts high and dives toward the asymptote).

Worked examples

Example 1: identify the parts (growth)

For $y = 3 \cdot 2^x$ , identify $a$ and $b$ , and decide growth or decay.

Match to the form

y = a \cdot b^x

The number out front is the initial value

a = 3

The base being raised to

x

b = 2

Since

b = 2 > 1

, the function grows

Answer: $a = 3$ , $b = 2$ — growth

Example 2: identify the parts (decay)

For $y = 5 \cdot \left(\dfrac{1}{2}\right)^x$ , identify $a$ and $b$ , and decide growth or decay.

The initial value is out front

a = 5

The base is the fraction

b = \dfrac{1}{2}

Since

0 < b < 1

, each step cuts the value in half — decay

Answer: $a = 5$ , $b = \dfrac{1}{2}$ — decay

Example 3: find the $y$ -intercept

Find the $y$ -intercept of $y = 7 \cdot 2^x$ .

Substitute

x = 0

y = 7 \cdot 2^0

Any nonzero base to the power

0

1

y = 7 \cdot 1

Simplify

y = 7

Answer: $(0, 7)$

Try one yourself

Common questions

How is exponential different from a quadratic like

y = x^2

In $y = x^2$ the variable is the base and the exponent is fixed. In an exponential like $y = 2^x$ the variable is the exponent. Quadratics make a U-shape; exponentials make a curve that races up or hugs the $x$ -axis.

What is a horizontal asymptote?

It is a horizontal line, $y = c$ , that the graph approaches but never touches. For $y = a \cdot b^x$ the asymptote is $y = 0$ — the outputs get tiny on one side but never reach zero.

Can

b

be negative?

No. In Algebra 1 the base must be positive (and not equal to $1$ ). A negative base would flip signs at every step and would not make a smooth curve.

What does

a

do to the graph?

It sets the starting height. $y = 5 \cdot 2^x$ and $y = 2^x$ have the same doubling behavior, but one crosses the $y$ -axis at $5$ and the other at $1$ .

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Introduction to Exponential Functions

The two parts: $a$ and $b$

Growth or decay: look at $b$

Exponential vs. linear

Worked examples

Example 1: identify the parts (growth)

Example 2: identify the parts (decay)

Example 3: find the $y$ -intercept

Try one yourself

Common questions

Keep learning

Graphing Exponential Functions

Writing Exponential Functions

Transformations of Exponential Functions

Geometric Sequences

Introduction to Exponential Functions

The two parts: aaa and bbb

Growth or decay: look at bbb

Exponential vs. linear

Worked examples

Example 1: identify the parts (growth)

Example 2: identify the parts (decay)

Example 3: find the yyy-intercept

Try one yourself

Common questions

Keep learning

Graphing Exponential Functions

Writing Exponential Functions

Transformations of Exponential Functions

Geometric Sequences

The two parts: $a$ and $b$

Growth or decay: look at $b$

Example 3: find the $y$ -intercept