Exponential functions

Variables in the exponent

An exponential function has the form

f(x) = a · bˣ

• a = initial value (the y-intercept)
• b = base (the growth or decay factor), b > 0, b ≠ 1

Growth and decay

Graph of y = 2ˣ

• Passes through (0, 1)
• Always positive
• Horizontal asymptote y = 0 (on the left)
• Domain: all real numbers; range: y > 0

y = (½)ˣ is the mirror image across the y-axis.

Exponential vs linear

Linear functions add the same amount; exponential functions multiply by the same factor. Exponential growth eventually beats any linear growth.

In a table: if y-values have a constant ratio, the function is exponential.

Growth and decay rates

Write b = 1 + r (growth) or b = 1 − r (decay).

• 5% growth: b = 1.05
• 12% decay: b = 0.88

Example: A population of 8,000 grows 3% per year: P = 8,000(1.03)ᵗ. After 10 years: ≈ 10,751.

Doubling and half-life

• Doubling: N = N₀(2)^(t/d), where d is the doubling time.
• Half-life: A = A₀(½)^(t/h), where h is the half-life.

Example: Carbon-14 has a half-life of about 5,730 years. After 11,460 years, ¼ remains.

Transformations

y = a · b^(x − h) + k shifts the horizontal asymptote to y = k.

The natural base e

e ≈ 2.718 appears in continuous growth: A = Pe^(rt). It’s used in science and finance.

Worked examples

Example 1: Is y = 3(0.6)ˣ growth or decay? Decay (b < 1); initial value 3.

Example 2: A $20,000 car loses 15% of its value each year. Value after 4 years? 20,000(0.85)⁴ ≈ $10,440.

Example 3: Does x: 0, 1, 2, 3; y: 5, 15, 45, 135 show exponential growth? Ratio 3 → yes, y = 5(3)ˣ.

Common mistakes

• Using 0.05 instead of 1.05 for 5% growth.
• Thinking exponential graphs touch the x-axis.
• Confusing constant differences (linear) with constant ratios (exponential).

Try it yourself

1. Initial value and base of y = 7(1.2)ˣ?
2. Growth or decay: y = 100(0.95)ˣ?
3. Write a model: 500 bacteria doubling each hour.
4. Horizontal asymptote of y = 2ˣ − 3?
5. $1,000 grows 6% per year. Value after 5 years?

Answers: 1) 7, 1.2 2) decay (5% per period) 3) N = 500(2)ᵗ 4) y = −3 5) ≈ $1,338.23

Quick recap

• Exponential functions: f(x) = a · bˣ.
• b > 1 is exponential growth; 0 < b < 1 is exponential decay.
• The graph has a horizontal asymptote; constant ratios signal exponentials.