Graphing Exponential Functions

Q: Which points should I put in my table?

Use $x$-values near zero: $-1, 0, 1, 2$ is usually enough. The point at $x = 0$ gives the $y$-intercept, and each step right just multiplies by the base.

Q: What are the domain and range?

The domain is all real numbers — any $x$ works. When $a > 0$, the range is $y > 0$: the outputs stay above the asymptote no matter what.

To graph an exponential function $y = a \cdot b^x$ , you only need a handful of points and one dashed line. The function multiplies by $b$ every time $x$ goes up by $1$ , so once you plot a few values the shape of the curve is locked in.

Two features anchor every exponential graph: the $y$ -intercept, which is always $(0, a)$ , and the horizontal asymptote $y = 0$ , the line the curve approaches but never touches. Find those first, fill in a short table, and connect the points with a smooth curve.

Start with the $y$ -intercept and the asymptote

Plug in $x = 0$ . Since $b^0 = 1$ , you get $y = a$ — the graph crosses the $y$ -axis at $(0, a)$ . That is the first point on every exponential graph and the fastest feature to spot.

Then draw a light dashed line at $y = 0$ . That is the horizontal asymptote. On one side of the graph the curve flattens toward this line; it never crosses it, because $a \cdot b^x$ is never zero.

Build a small table

Pick easy $x$ -values around zero, like $-1, 0, 1, 2, 3$ . Moving one step right multiplies $y$ by $b$ ; moving one step left divides by $b$ . For $y = 2^x$ the table gives $\dfrac{1}{2}, 1, 2, 4, 8$ — each entry double the one before.

Plot the points and connect them with a smooth curve that hugs the asymptote on one end and climbs steeply on the other. Do not connect them with straight segments — the curve bends continuously.

Reading direction: growth vs. decay

Read the graph left to right. If $b > 1$ , the curve rises — growth. If $0 < b < 1$ , the curve falls toward the asymptote — decay. Either way the domain is all real numbers, and the range is $y > 0$ whenever $a$ is positive.

Worked examples

Example 1: graph $y = 2^x$ from a table

Make a table for $y = 2^x$ and identify the asymptote.

Evaluate at

x = -1

y = 2^{-1} = \dfrac{1}{2}

Evaluate at

x = 0

y = 2^0 = 1

Evaluate at

x = 1, 2, 3

y = 2,\ 4,\ 8

The curve flattens toward the line

y = 0

Answer: Points $\left(-1, \dfrac{1}{2}\right), (0, 1), (1, 2), (2, 4), (3, 8)$ ; asymptote $y = 0$

Example 2: find the $y$ -intercept

Find the $y$ -intercept of $y = 3 \cdot 2^x$ .

Substitute

x = 0

y = 3 \cdot 2^0

Apply

2^0 = 1

y = 3

Answer: $(0, 3)$

Example 3: a decay curve

Describe the graph of $y = 5 \cdot \left(\dfrac{1}{2}\right)^x$ .

The

y

-intercept is the initial value

(0, 5)

Each step right halves the output

5,\ 2.5,\ 1.25,\ \ldots

The curve falls left to right toward

y = 0

Answer: A falling curve through $(0, 5)$ with asymptote $y = 0$

Try one yourself

Common questions

Does an exponential graph ever touch the

x

-axis?

No. The outputs of $y = a \cdot b^x$ get very close to $0$ but never equal $0$ , so the curve approaches the line $y = 0$ without touching it. That line is the horizontal asymptote.

Which points should I put in my table?

Use $x$ -values near zero: $-1, 0, 1, 2$ is usually enough. The point at $x = 0$ gives the $y$ -intercept, and each step right just multiplies by the base.

How do I tell growth from decay on a graph?

Read left to right. Rising means the base is greater than $1$ (growth); falling toward the asymptote means the base is between $0$ and $1$ (decay).

What are the domain and range?

The domain is all real numbers — any $x$ works. When $a > 0$ , the range is $y > 0$ : the outputs stay above the asymptote no matter what.

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Graphing Exponential Functions

Start with the $y$ -intercept and the asymptote

Build a small table

Reading direction: growth vs. decay

Worked examples

Example 1: graph $y = 2^x$ from a table

Example 2: find the $y$ -intercept

Example 3: a decay curve

Try one yourself

Common questions

Keep learning

Introduction to Exponential Functions

Transformations of Exponential Functions

Writing Exponential Functions

Graphing Exponential Functions

Start with the yyy-intercept and the asymptote

Build a small table

Reading direction: growth vs. decay

Worked examples

Example 1: graph y=2xy = 2^xy=2x from a table

Example 2: find the yyy-intercept

Example 3: a decay curve

Try one yourself

Common questions

Keep learning

Introduction to Exponential Functions

Transformations of Exponential Functions

Writing Exponential Functions

Start with the $y$ -intercept and the asymptote

Example 1: graph $y = 2^x$ from a table

Example 2: find the $y$ -intercept