What Is an Exponential Function?

An exponential function has the form y = a × b^x, where the variable x is in the exponent. The base b determines whether the function models growth or decay.

General Form

y = a × b^x

a = initial value (y-intercept)

b = base (growth/decay factor)

x = time or independent variable

Exponential Growth (b > 1)

When b > 1, the function increases rapidly. Example: y = 2^x doubles for every increase of 1 in x.

Exponential Decay (0 < b < 1)

When 0 < b < 1, the function decreases towards zero. Example: y = (0.5)^x halves for every increase of 1 in x.

Key Graph Features

All exponential functions y = a × b^x (with a > 0) share these key features:

y-intercept

(0, a)

Always passes through (0, a)

Horizontal Asymptote

y = 0

Approaches but never reaches the x-axis

Domain & Range

All x; y > 0

Defined for all x, always positive

Growth: y = 2^x

Decay: y = (0.5)^x

Growth and Decay Applications

Exponential functions model many real-world situations. The growth or decay rate r is often expressed as a percentage.

Growth/Decay Formula

A = P(1 + r)^t (growth) or A = P(1 − r)^t (decay)

P = initial amount, r = rate (decimal), t = time

1

Population Growth

A town has 5000 people and grows at 3% per year. How many people after 10 years?

P = 5000, r = 0.03, t = 10

A = 5000 × (1.03)¹⁰ = 5000 × 1.3439 ≈ 6720 people

2

Radioactive Decay

A 200 g sample decays at 5% per year. How much remains after 8 years?

P = 200, r = 0.05, t = 8

A = 200 × (0.95)⁸ = 200 × 0.6634 ≈ 132.7 g

3

Halving Time

A substance has a half-life of 10 years. Starting with 80 g, how much remains after 30 years?

After 10 years: 80 × ½ = 40 g

After 20 years: 40 × ½ = 20 g

After 30 years: 20 × ½ = 10 g

Alternatively: A = 80 × (0.5)³ = 80/8 = 10 g

Key Vocabulary

Asymptote

A line that a graph approaches but never actually reaches. Exponential graphs have a horizontal asymptote at y = 0.

Growth Factor

The base b in y = a × b^x. If b > 1, the quantity grows; if 0 < b < 1, it decays.

Half-Life

The time it takes for a quantity undergoing exponential decay to reduce to half its initial value.

Exponential Decay

A decrease modelled by y = a × b^x with 0 < b < 1. The quantity decreases rapidly at first, then more slowly.

Knowledge Check

Test your understanding of exponential functions and their applications.

Question 1

For y = 3 × 2^x, what is the y-intercept?

2 6 1 3

Question 2

A population of 1000 bacteria doubles every hour. How many are there after 3 hours?

8000 6000 3000 2000

Question 3

Which function represents exponential decay?

y = 5 × 2^x y = 4 × (0.8)^x y = 3^x y = 2 × 1.5^x

Question 4

An investment of $2000 grows at 4% per year. What is its value after 5 years? (to the nearest dollar)

$2400 $2500 $2433 $2480

Question 5

What is the horizontal asymptote of y = 5 × 3^x?

y = 5 y = 3 x = 0 y = 0

Exponential Functions