🎯 Learning Objectives
- What an exponential function looks like and how it differs from a linear or quadratic function.
- Growth vs. decay: how to spot each from the equation.
- Compound interest and population formulas you must know.
- How to solve simple exponential equations.
- The 7 essential exponent rules.
- Practice questions with worked solutions.
1. What Is an Exponential Function?
Exponential form. b > 0 and b ≠ 1.
y = a · b^x
In an exponential function, the variable lives in the EXPONENT: not in the base. The constant b is called the base.
2. Growth or Decay? Check the Base
❓ What is the base b? | ||
b > 1 ▼ | 0 < b < 1 ▼ | |
GROWTH: values increase rapidly | DECAY: values decrease toward zero | |
Equation | Type | Reason |
y = 100·(1.05)ˣ | Growth | 1.05 > 1 (5% growth) |
y = 80·(0.92)ˣ | Decay | 0.92 < 1 (8% loss) |
y = 50·(2)ˣ | Growth (doubling) | Base 2 > 1 |
y = 200·(1/2)ˣ | Decay (halving) | Base 1/2 < 1 |
3. Real-World Models
Compound growth (P initial, r rate, t years)
A = P · (1 + r)^t
Compound decay
A = P · (1 − r)^t
Compound interest n times per year
A = P · (1 + r/n)^(n·t)
4. The 7 Exponent Rules
Rule | Example |
x^a · x^b = x^(a+b) | x³ · x² = x⁵ |
x^a / x^b = x^(a−b) | x⁵ / x² = x³ |
(x^a)^b = x^(a·b) | (x²)³ = x⁶ |
(x·y)^a = x^a · y^a | (2x)³ = 8x³ |
x⁰ = 1 (when x ≠ 0) | 5⁰ = 1 |
x^(−a) = 1 / x^a | x⁻² = 1/x² |
x^(1/n) = ⁿ√x | x^(1/2) = √x |
5. Solving Exponential Equations
💡 Same Base Strategy
If both sides can be written with the SAME base, set the exponents equal. Example: 2^x = 8 → 2^x = 2³ → x = 3.
📝 Example: Solve 9^x = 27
Same base: 9 = 3² and 27 = 3³.
Rewrite: (3²)^x = 3³ → 3^(2x) = 3³
Set exponents equal: 2x = 3 → x = 3/2
6. Strategies
⚡ STRATEGY 1: Identify Growth or Decay
The base is the only thing that matters. > 1 = growth. < 1 = decay. Done.
⚡ STRATEGY 2: Convert percentages to multipliers
5% growth → multiplier 1.05. 8% decay → multiplier 0.92. 30% growth → 1.30. This conversion saves time on word problems.
⚡ STRATEGY 3: y-intercept of an exponential
Plug x = 0. Since b⁰ = 1, the y-intercept is just a (the leading coefficient).
⚠️ Common Mistakes
- Confusing 1.05 (growth) with 0.05 (the rate, not the multiplier).
- Forgetting to convert percentage to a decimal before adding to 1.
- Computing 2^x as 2x: the variable is the EXPONENT, not a coefficient.
- Saying decay means negative numbers: decay means values shrink toward zero, not below it.
7. Summary
📌 Key Takeaways
- Form: y = a · b^x
- Growth: b > 1
- Decay: 0 < b < 1
- Compound growth: A = P(1 + r)^t
- y-intercept: y = a (since b⁰ = 1)
- Same-base strategy: Rewrite both sides with same base; set exponents equal.
8. Practice: 10 Questions
Q1. A population of 500 grows by 3% per year. After t years it equals…
A) 500·(0.03)^t B) 500·(1.03)^t C) 500 + 0.03t D) 500·(3)^t
Hint: Growth multiplier = 1.03.
Q2. The function y = 200·(0.85)^x represents…
A) Growth B) Decay C) Constant D) Linear
Hint: Base 0.85 < 1 → decay.
Q3. Solve: 2^x = 32.
A) 4 B) 5 C) 6 D) 16
Hint: 32 = 2⁵.
Q4. Solve: 3^(x+1) = 27.
A) 1 B) 2 C) 3 D) 4
Hint: 27 = 3³, so x + 1 = 3 → x = 2.
Q5. Simplify: x⁵ · x³.
A) x⁸ B) x¹⁵ C) x² D) 2x⁸
Hint: Same base, add exponents.
Q6. Simplify: (x²)³.
A) x⁵ B) x⁶ C) x⁸ D) x⁹
Hint: Multiply exponents.
Q7. A car valued at $30,000 loses 12% of its value per year. Value after t years…
A) 30000·(0.12)^t B) 30000·(0.88)^t C) 30000·(1.12)^t D) 30000 − 12t
Hint: Decay multiplier = 1 − 0.12 = 0.88.
Q8. The y-intercept of y = 6·(2)^x is…
A) 0 B) 2 C) 6 D) 12
Hint: Plug x = 0: y = 6·1 = 6.
Q9. Simplify: 8^(2/3).
A) 2 B) 4 C) 8 D) 16
Hint: 8^(1/3) = 2, then 2² = 4.
Q10. A bacteria culture doubles every hour, starting with 50 bacteria. After t hours…
A) 50 + 2t B) 50·t² C) 50·(2)^t D) 50·(1.02)^t
Hint: Doubling = base 2.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | B) 500·(1.03)^t | Growth → 1 + r. |
Q2. | B) Decay | Base < 1. |
Q3. | B) 5 | 32 = 2⁵. |
Q4. | B) 2 | x + 1 = 3. |
Q5. | A) x⁸ | Add exponents. |
Q6. | B) x⁶ | Power-of-power: multiply. |
Q7. | B) 30000·(0.88)^t | Decay → 1 − r. |
Q8. | C) 6 | y-int = leading coefficient. |
Q9. | B) 4 | (8^(1/3))² = 2² = 4. |
Q10. | C) 50·(2)^t | Doubling = base 2. |