What is an Exponential Function?

An exponential function has the variable in the exponent. The general form is:

y = a · b^x

where a ≠ 0, b > 0, and b ≠ 1

The value a is the initial value (the y-intercept when x = 0, since b⁰ = 1). The value b is the base, which determines growth or decay:

Growth: b > 1

The function increases rapidly. Example: population growth, compound interest.

Decay: 0 < b < 1

The function decreases toward zero. Example: radioactive decay, depreciation.

Exponential Curves

Growth (b > 1)

e.g. y = 2^x

Decay (0 < b < 1)

e.g. y = (1/2)^x

Transformations of Exponential Functions

The general transformed exponential function is:

y = a · b^{(x - h)} + k

Parameter	Effect
a	Vertical stretch/compression. If a < 0, reflects across x-axis.
h	Horizontal shift: right if h > 0, left if h < 0.
k	Vertical shift: up if k > 0, down if k < 0. The horizontal asymptote becomes y = k.

The horizontal asymptote of y = a · b^(x-h) + k is y = k. The basic exponential y = b^x has asymptote y = 0.

Applications

Exponential functions model many real-world phenomena:

Compound Interest

A = P(1 + r/n)^nt

P = principal, r = rate, n = compounds per year, t = years

Population Growth

P(t) = P₀ · e^kt

P₀ = initial population, k = growth rate, t = time

Radioactive Decay

A(t) = A₀ · (1/2)^t/h

A₀ = initial amount, h = half-life, t = time

Depreciation

V(t) = V₀ · (1 - r)^t

V₀ = initial value, r = rate of depreciation, t = years

Key Vocabulary

Exponential Growth

Rapid increase where the rate of growth is proportional to the current value. The base b > 1.

Exponential Decay

Gradual decrease toward zero where 0 < b < 1. The function never reaches zero.

Asymptote

A line that the graph approaches but never reaches. For y = b^x, the asymptote is y = 0.

Base (b)

The constant being raised to the power of x. Must be positive and not equal to 1.

Worked Examples

1

Sketch y = 2^x and state the domain, range, and asymptote.

Step 1: Key points: (0, 1), (1, 2), (2, 4), (-1, 0.5), (-2, 0.25).

Step 2: The curve rises steeply to the right and approaches 0 to the left.

Answer: Domain = (-∞, ∞). Range = (0, ∞). Asymptote: y = 0. Y-intercept: (0, 1).

2

$5000 is invested at 6% p.a. compound interest. Find the value after 10 years.

Step 1: Use A = P(1 + r)^t with P = 5000, r = 0.06, t = 10.

Step 2: A = 5000 × (1.06)¹⁰ = 5000 × 1.7908...

Answer: A ≈ $8954.24.

3

State the asymptote and range of y = 3^x - 2.

Step 1: This is y = 3^x shifted down by 2 units (k = -2).

Step 2: The asymptote shifts from y = 0 to y = -2.

Answer: Asymptote: y = -2. Range = (-2, ∞).

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

What is the y-intercept of y = 5 · 3^x?

5 3 15 1

Question 2

The function y = (0.5)^x represents:

Exponential growth Exponential decay Linear growth Quadratic growth

Question 3

What is the horizontal asymptote of y = 2^x + 5?

y = 0 y = 2 y = 5 y = -5

Question 4

$1000 is invested at 5% p.a. compound interest. After 4 years, the approximate value is:

$1200 $1500 $1050 $1215.51

Question 5

What is the range of y = -2^x?

(-∞, 0) (0, ∞) (-∞, ∞) [-2, ∞)

Exponential Functions