4.1Understanding Exponential Growth and Decay
Define f(x) = bˣ, distinguish exponential growth (b > 1) from decay (0 < b < 1), evaluate the natural base e, and apply compound interest A = P(1 + r/n)^(nt).
Exponential functions model population growth, radioactive decay, compound interest, and viral spread. They are among the most important functions in mathematics, science, and finance — mastering them unlocks calculus, statistics, and real-world modeling.
Essential Question
How does an exponential function differ from a polynomial function, and what real-world phenomena does each type best model?
Lesson Overview
An exponential function has the form f(x) = ab^x where b > 0 and b ≠ 1. Unlike polynomial functions where x is the base, here x is the exponent. This single difference produces dramatically different behavior: exponential functions grow (or decay) at a rate proportional to their current value.
Key Formula
f(x) = ab^x
- a = initial value (y-intercept when x = 0)
- b = base (growth/decay factor)
- x = exponent (independent variable)
Growth vs. Decay
- b > 1: function grows as x increases
- 0 < b < 1: function decays as x increases
- b = 1: constant function (not exponential)
- b ≤ 0: not a valid exponential base
f(x) = 2ˣ — exponential growth with base 2
Growth (b > 1) vs. Decay (0 < b < 1)
Worked Examples
Evaluate f(x) = 3 · 2ˣ at x = 0, x = 3, and x = −2.
f(0) = 3 · 2⁰ = 3 · 1 = 3
f(3) = 3 · 2³ = 3 · 8 = 24
f(−2) = 3 · 2⁻² = 3 · (1/4) = 3/4 = 0.75
Pattern: each time x increases by 1, the output doubles (multiplied by base 2).
Classify each as growth or decay: (a) f(x) = 5·(1.3)ˣ (b) g(x) = 100·(0.85)ˣ (c) h(x) = 4·(2/3)ˣ
(a) Base = 1.3 > 1 → GROWTH. The function increases as x increases.
(b) Base = 0.85, and 0 < 0.85 < 1 → DECAY. The function decreases as x increases.
(c) Base = 2/3 ≈ 0.667, and 0 < 2/3 < 1 → DECAY.
Key check: look only at the base b. If b > 1: growth. If 0 < b < 1: decay.
Evaluate the natural exponential function f(x) = eˣ at x = 0, 1, 2, and −1.
e ≈ 2.71828 is the natural base. It arises in continuous growth and calculus.
f(0) = e⁰ = 1
f(1) = e¹ ≈ 2.718
f(2) = e² ≈ 7.389
f(−1) = e⁻¹ ≈ 0.368
Continuous growth formula: A = Pe^(rt), where P = principal, r = rate, t = time.
$5000 is invested at 4% annual rate, compounded monthly, for 3 years. Find the final amount.
Formula: A = P(1 + r/n)^(nt)
Identify: P = 5000, r = 0.04, n = 12 (monthly), t = 3
Substitute: A = 5000(1 + 0.04/12)^(12·3)
= 5000(1.003333...)^36
= 5000(1.12749...)
An exponential function passes through (0, 4) and (2, 36). Find f(x) = ab^x.
Use (0, 4): f(0) = a · b⁰ = a · 1 = a = 4.
Use (2, 36): f(2) = 4 · b² = 36, so b² = 9, thus b = 3 (b must be positive).
Answer: f(x) = 4 · 3ˣ
Verify: f(0) = 4·1 = 4 ✓ f(2) = 4·9 = 36 ✓
Guided Practice
Evaluate g(x) = 2 · 5ˣ at x = 0, x = 2, and x = −1.
Hint: Apply exponent rules: 5⁰ = 1, 5² = 25, 5⁻¹ = 1/5.
Classify each as growth or decay: (a) f(x) = 7·(1.05)ˣ (b) h(x) = 3·(0.4)ˣ
Hint: Compare the base to 1. If b > 1: growth. If 0 < b < 1: decay.
Find the y-intercept of f(x) = 6·(1.2)ˣ and state whether it grows or decays.
Hint: The y-intercept is f(0). Recall b⁰ = 1 for any valid base.
$2000 is invested at 5% annual rate compounded quarterly for 4 years. Find the final amount.
Hint: Use A = P(1 + r/n)^(nt) with n = 4 (quarterly).
An exponential function passes through (0, 10) and (1, 30). Write f(x) = ab^x.
Hint: Use (0, 10) to find a first, then use (1, 30) to find b.
Key Vocabulary
Exponential Function
A function of the form f(x) = ab^x where b > 0 and b ≠ 1; the variable x appears as the exponent.
Base (b)
The constant multiplied repeatedly in an exponential function. If b > 1: growth; if 0 < b < 1: decay.
Exponential Growth
When b > 1, the function increases without bound as x → +∞ and approaches 0 as x → −∞.
Exponential Decay
When 0 < b < 1, the function decreases toward 0 as x → +∞ and grows without bound as x → −∞.
Natural Base e
The irrational number e ≈ 2.71828, the base of the natural exponential function f(x) = eˣ.
Compound Interest
Interest calculated on both principal and previously earned interest. Formula: A = P(1 + r/n)^(nt).
Horizontal Asymptote
A horizontal line the graph approaches but never crosses. For f(x) = ab^x, the asymptote is y = 0.
Initial Value (a)
The value of f(0) = a in f(x) = ab^x; the y-intercept of the exponential function.
Check Your Understanding
Interactive Practice — 5 Questions
Which of the following is an exponential function?
For f(x) = 5·(0.7)ˣ, what is the y-intercept and does it grow or decay?
Evaluate f(x) = 4·3ˣ at x = −2.
$1000 is compounded annually at 8% for 2 years. What is the final amount?
An exponential function passes through (0, 2) and (3, 54). What is f(x)?
Independent Practice
Independent Practice
Evaluate f(x) = 4·2ˣ at x = −3, x = 0, and x = 4.
Classify each as growth or decay and state the y-intercept: (a) f(x) = 0.5·(3)ˣ (b) g(x) = 8·(0.25)ˣ
Write the exponential function f(x) = ab^x that passes through (0, 7) and (2, 63).
$3500 is invested at 6% annual rate compounded semi-annually for 5 years. Find the final amount.
A population of bacteria starts at 200 and triples every hour. Write f(t) and find the population after 4 hours.
Common Mistakes
Confusing f(x) = x² (power function) with f(x) = 2ˣ (exponential function).
In a power function, x is the base. In an exponential function, x is the exponent. They grow at very different rates.
Using a negative or zero base: f(x) = (−2)ˣ or f(x) = 0ˣ.
The base b must satisfy b > 0 and b ≠ 1. Negative bases cause oscillating signs; b = 0 or b = 1 are degenerate cases.
Thinking the y-intercept is (0, b) instead of (0, a).
f(0) = a·b⁰ = a·1 = a. The base b does not affect the y-intercept — only a does.
Using the interest rate as a percent instead of a decimal: A = P(1 + 6/n)^(nt).
In A = P(1 + r/n)^(nt), r must be a decimal. Convert 6% → r = 0.06 before substituting.
Math Tips
To quickly identify growth vs. decay: if the base b > 1, the function grows; if 0 < b < 1, it decays.
The y-intercept of f(x) = ab^x is always (0, a) — just substitute x = 0.
For compound interest: "monthly" means n = 12; "quarterly" means n = 4; "daily" means n = 365.
e ≈ 2.718 is between 2 and 3, so eˣ grows faster than 2ˣ but slower than 3ˣ.
When writing an exponential from two points, always find a first using the point where x = 0.