In a linear function, the output changes by a constant amount for every unit increase in x (you add or subtract the same number). In an exponential function, the output changes by a constant factor (you multiply or divide by the same number).
Standard form: y = a · bˣ
Example: A population of 500 bacteria doubles every hour.
Reading the equation:
Example: A radioactive substance starts at 200 grams and loses 10% of its mass each year.
After 3 years: M = 200 · (0.90)³ = 200 · 0.729 = 145.8 grams
The relationship between percent change and the growth factor is critical:
| Situation | Growth Factor (b) | |---|---| | Increases by 15% per year | b = 1 + 0.15 = 1.15 | | Increases by 100% (doubles) | b = 1 + 1.00 = 2 | | Decreases by 20% per year | b = 1 − 0.20 = 0.80 | | Decreases by 50% per period | b = 0.50 |
The initial value a corresponds to b⁰ = 1 — it's the value at the start.
> Q: "The equation V = 18,000 · (0.85)ᵗ models the value V of a car t years after purchase. What does 0.85 represent?"
The answer: The car retains 85% of its value each year — equivalently, it loses 15% of its value per year.
> Q: "What does 18,000 represent?"
The initial purchase price (value when t = 0).
Real-world example: A social media post gets 3 times as many views each day after it goes viral. On Day 0, it had 400 views. Write the equation and find views after 4 days.
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Quiz Questions:
Q1. A savings account starts with $1,000 and earns 5% annual interest, compounded annually. Which equation models the account balance B after t years?
A) B = 1,000 + 0.05t B) B = 1,000 · (1.05)ᵗ C) B = 1,000 · (0.05)ᵗ D) B = 1,000 · (0.95)ᵗ
Answer: B — This is exponential growth. Initial value = 1,000. Growth factor = 1 + 0.05 = 1.05 (earning 5% per year means the balance is multiplied by 1.05 each year). Choice A is linear, not exponential. Choice C uses the interest rate itself as the base, which is wrong. Choice D would be decay (losing 5% per year).
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Q2. The equation N = 500 · (0.75)ᵈ models the number of bacteria N remaining in a sample after d days of treatment. What does 0.75 represent?
A) 75% of bacteria are eliminated each day B) The sample retains 75% of its bacteria each day (decreasing by 25% per day) C) The sample starts with 75 bacteria D) After 0.75 days, the sample is eliminated
Answer: B — 0.75 is the decay factor. Each day, 75% of bacteria survive (and 25% are eliminated). Choice A incorrectly says 75% are eliminated; only 25% are. Choice C confuses the growth factor with the initial value.
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Q3. A car's value decreases by 12% each year. If the car originally cost $24,000, which equation models its value V after t years?
A) V = 24,000 − 0.12t B) V = 24,000 · (0.12)ᵗ C) V = 24,000 · (0.88)ᵗ D) V = 24,000 · (1.12)ᵗ
Answer: C — Depreciation of 12% per year means the car retains 88% of its value annually. Decay factor = 1 − 0.12 = 0.88. So V = 24,000 · (0.88)ᵗ. Choice A is linear. Choice B uses 0.12 as the factor (wrong). Choice D would mean the car grows in value by 12% per year.
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Q4. A quantity doubles every 3 years. If the initial amount is 50, what is the amount after 9 years?
A) 200 B) 300 C) 400 D) 600
Answer: C — Doubling every 3 years: after 3 years → 100; after 6 years → 200; after 9 years → 400. Using the formula: 50 · 2^(9/3) = 50 · 2³ = 50 · 8 = 400.
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Q5. An exponential function passes through (0, 4) and (1, 12). What is the growth factor?
A) 8 B) 3 C) 4 D) 12
Answer: B — The initial value a = 4 (when x = 0). When x = 1: 12 = 4 · b¹ → b = 12/4 = 3. The growth factor is 3 (the quantity triples each period).