The output is always positive.

It can become extremely small:

0.1

0.01

0.001

and so on.

But it never becomes exactly zero.

It also never becomes negative.

Therefore, for y = 2ˣ:

Range: y > 0

or in interval notation:

(0, ∞)

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Horizontal Asymptotes

Look carefully at what happens to:

y = 2ˣ

as x becomes increasingly negative.

For example:

2⁻¹ = 0.5

2⁻² = 0.25

2⁻³ = 0.125

2⁻⁴ = 0.0625

2⁻⁵ = 0.03125

The values get closer and closer to zero.

However, 2ˣ never actually equals zero.

The line:

y = 0

is therefore a horizontal asymptote.

A horizontal asymptote is a horizontal line that a graph approaches as x moves far in one direction.

For a basic exponential function:

y = a(bˣ)

the horizontal asymptote is usually:

y = 0

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Does an Exponential Graph Touch Its Asymptote?

For a basic positive exponential function, no.

For example:

y = 3ˣ

can become extremely close to zero, but:

3ˣ > 0

for every real value of x.

Therefore, its graph never reaches the x-axis.

This is why the x-axis acts as the horizontal asymptote.

An asymptote describes the long-term behaviour of the graph.

---

Shifting an Exponential Graph Vertically

Now consider:

y = 2ˣ + 3

Adding 3 moves the entire graph 3 units upward.

The original function:

y = 2ˣ

has horizontal asymptote:

y = 0

The transformed function:

y = 2ˣ + 3

has horizontal asymptote:

y = 3

Its range also changes.

Instead of:

y > 0

the range becomes:

y > 3

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A Useful General Form

A transformed exponential function can be written as:

y = a(bˣ) + k

The value k moves the graph vertically.

Its horizontal asymptote becomes:

y = k

For example:

y = 4(2ˣ) + 5

has horizontal asymptote:

y = 5

and, because the graph is above the asymptote:

Range: y > 5

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A Reflection of an Exponential Graph

A negative coefficient can reflect an exponential graph across the x-axis.

Compare:

y = 2ˣ

and:

y = −2ˣ

For y = −2ˣ:

• all outputs are negative
• the horizontal asymptote is still y = 0
• the domain is all real numbers
• the range is y < 0

This reminds us that the range is not always y > 0. We must examine the complete function.

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Worked Example 1: Sketching an Exponential Growth Function

Sketch:

y = 2(3ˣ)

Step 1: Identify the base

The base is:

b = 3

Since:

3 > 1

the function represents exponential growth.

Step 2: Find some points

Step 3: Identify the asymptote

There is no vertical shift, so:

Horizontal asymptote: y = 0

Step 4: State the domain and range

Domain: all real numbers

Range: y > 0

Step 5: Sketch the curve

The graph rises increasingly rapidly from left to right.

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Worked Example 2: Exponential Decay

Consider:

y = 8(0.5ˣ)

Because:

0 < 0.5 < 1

the function represents exponential decay.

Calculate some values:

From the graph:

Domain: all real numbers

Range: y > 0

Horizontal asymptote: y = 0

Behaviour: y decreases toward zero as x increases.

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Worked Example 3: Finding a Shifted Asymptote

Consider:

y = 3(2ˣ) − 4

The −4 shifts the basic exponential graph downward by 4 units.

Therefore:

Horizontal asymptote: y = −4

Because 3(2ˣ) is always positive, the graph remains above −4.

Therefore:

Domain: all real numbers

Range: y > −4

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Interpreting Exponential Graphs in Context

Exponential functions are useful because many real-world quantities change by a constant percentage or constant multiplication factor.

Examples include:

• population growth
• bacterial growth
• compound interest
• radioactive decay
• depreciation
• cooling models
• spread of information
• some biological growth processes

The graph can tell us much more than whether a quantity is increasing or decreasing.

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Example: Bacterial Growth

Suppose a bacterial population is modelled by:

P = 200(2ᵗ)

where:

• P is the bacterial population
• t is time in hours

At t = 0:

P = 200(2⁰)

P = 200

So the initial population is 200 bacteria.

After 1 hour:

P = 200(2¹) = 400

After 2 hours:

P = 200(2²) = 800

After 3 hours:

P = 200(2³) = 1600

The graph becomes increasingly steep because the population doubles during every equal time interval.

In the real-world context, we would normally restrict the domain to:

t ≥ 0

because negative time may not make sense for the situation being modelled.

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Example: Depreciation

Suppose a piece of equipment is initially worth $10,000 and retains 80% of its value each year.

Its value could be modelled by:

V = 10,000(0.8ᵗ)

Because:

0 < 0.8 < 1

this represents exponential decay.

After 1 year:

V = 10,000(0.8)

V = $8,000

After 2 years:

V = 10,000(0.8²)

V = $6,400

After 3 years:

V = 10,000(0.8³)

V = $5,120

The graph approaches zero but does not reach zero according to the mathematical model.

In reality, an object might eventually have a minimum resale or scrap value, so mathematical models do not always describe reality perfectly forever.

---

Reading an Exponential Graph

When interpreting an exponential graph, ask several questions.

1. What does the horizontal axis represent?

It might represent:

• time
• distance
• number of periods
• age

2. What does the vertical axis represent?

It might represent:

• population
• money
• mass
• concentration
• temperature difference

3. What is the initial value?

Often this is the value when:

x = 0

4. Is the quantity growing or decaying?

Look at whether the graph rises or falls from left to right.

5. What value does the graph approach?

This can often be identified from the horizontal asymptote.

6. What restrictions make sense?

A mathematical function might allow all real x-values, but the real-world situation may not.

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Linear Growth vs Exponential Growth

These two types of growth are easily confused.

Consider:

Linear: y = 2x + 1

Exponential: y = 2ˣ

A linear function changes by a constant amount.

An exponential function changes by a constant factor or percentage.

For example:

The exponential function eventually grows much faster.

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Common Mistakes

Mistake 1: Thinking exponential growth means constant addition

The sequence:

2, 4, 6, 8, 10

is not exponential.

It increases by 2 each time, so it shows a linear pattern.

The sequence:

2, 4, 8, 16, 32

is exponential because each term is multiplied by 2.

---

Mistake 2: Thinking an exponential graph eventually reaches its asymptote

For:

y = 2ˣ

the graph approaches y = 0 but never reaches it.

The x-axis is a horizontal asymptote.

---

Mistake 3: Assuming all exponential graphs increase

Functions with:

b > 1

show exponential growth.

Functions with:

0 < b < 1

show exponential decay.

---

Mistake 4: Saying the range is always y > 0

That is true for basic positive exponential functions such as:

y = 2ˣ

But transformations can change the range.

For:

y = 2ˣ + 5

the range is:

y > 5

---

Mistake 5: Confusing x² with 2ˣ

These functions are fundamentally different.

x² → quadratic

2ˣ → exponential

The position of the variable matters.

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Key Terms

Exponential function — A function in which the variable occurs in the exponent.

Base — The number being raised to a power.

Exponential growth — An exponential pattern in which values increase by a constant multiplication factor.

Exponential decay — An exponential pattern in which values decrease by a constant multiplication factor between 0 and 1.

Domain — The set of possible input values.

Range — The set of possible output values.

Horizontal asymptote — A horizontal line that a graph approaches as x moves far in one direction.

Initial value — The output at the beginning of a model, commonly when x = 0.

Growth factor — The factor greater than 1 by which a quantity is repeatedly multiplied.

Decay factor — A factor between 0 and 1 by which a quantity is repeatedly multiplied.

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Check Your Understanding

1. Is y = 4ˣ an exponential growth or decay function?

2. Is y = (0.3)ˣ growth or decay?

3. What point must the graph of y = 7ˣ pass through?

4. State the domain of y = 5ˣ.

5. State the range of y = 5ˣ.

6. What is the horizontal asymptote of y = 5ˣ?

7. What is the horizontal asymptote of y = 5ˣ + 4?

8. State the range of y = 5ˣ + 4.

9. Create a table of values for y = 2ˣ using x = −2, −1, 0, 1, 2 and sketch the graph.

10. A population is modelled by P = 500(1.2ᵗ). Is the population growing or decaying? Explain how you know.

11. A machine's value is modelled by V = 20,000(0.9ᵗ). What does 20,000 represent? What does 0.9 tell you about the machine's value?

12. Explain why an exponential decay graph can approach zero indefinitely without actually reaching zero.

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Key Takeaways

• An exponential function has the variable in the exponent.
• A basic exponential function can be written as y = a(bˣ).
• If b > 1, the function shows exponential growth.
• If 0 < b < 1, the function shows exponential decay.
• Basic exponential functions have a domain of all real numbers.
• A basic positive exponential function has the range y > 0.
• The basic exponential graph has the horizontal asymptote y = 0.
• In y = a(bˣ) + k, the horizontal asymptote is y = k.
• Exponential change involves repeated multiplication, not repeated addition.
• Graphs should always be interpreted using the meaning of their axes and the context of the situation.

Did You Know?

Exponential growth can become surprisingly large very quickly. If a quantity doubles repeatedly, after only 10 doublings it is 1,024 times its original size. After 20 doublings, it is more than one million times its original size. This rapid change is one reason exponential models are so important in areas such as finance, biology, computing, and population studies.