Topic 4 of 7
Exponents and exponential growth
Step 11 · Algebra 1 topics
Model growth and decay that multiply by the same factor each period, use compound interest and half-life, and tell linear from exponential patterns.
Growth by a factor
Exponential growth multiplies by the same factor in each time period. f(t) = a · bᵗ starts at a and is multiplied by b each step. For growth, b is bigger than 1.
| Day | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Bacteria | 100 | 200 | 400 | 800 | 1,600 |
Example: A growth rate
A town of 20,000 grows 3% a year. Write a function and find the population after 2 years.
- Each year multiplies by 1.03: P(t) = 20,000 · 1.03ᵗ.
- P(2) = 20,000 · 1.03², and 1.03² = 1.0609.
- 20,000 × 1.0609 = 21,218.
Answer21,218 people
Example: Compound interest
$1,000 earns 5% interest, compounded once a year. How much is there after 3 years?
- Multiply by 1.05 three times: 1,000 · 1.05³.
- 1.05³ is 1.157625.
- 1,000 × 1.157625 is 1,157.625, which is $1,157.63 to the cent.
Answer$1,157.63
Exponential decay
Exponential decay multiplies by a factor between 0 and 1. A decay rate r means a factor of 1 − r.
Example: Losing value
A $24,000 car loses 15% of its value each year. What is it worth after 2 years?
- The factor is 1 − 0.15 = 0.85.
- 0.85² = 0.7225.
- 24,000 × 0.7225 = 17,340.
Answer$17,340
Example: Half-life
The amount of a medicine in the body halves every 6 hours. A 400 mg dose is taken. How much is left after 18 hours?
- 18 hours is 3 half-lives.
- 400, then 200, then 100, then 50.
Answer50 mg
Linear or exponential?
Linear functions add the same amount each step; exponential functions multiply by the same factor. In the long run, exponential growth always overtakes linear growth.
| x | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| 3x + 10 | 10 | 13 | 16 | 19 | 22 | 25 |
| 2ˣ | 1 | 2 | 4 | 8 | 16 | 32 |
Example: Which model?
A table shows y = 5, 15, 45, 135 for x = 0, 1, 2, 3. Is it linear or exponential?
- The differences are 10, 30 and 90: not constant.
- The ratios are 3, 3 and 3: constant.
AnswerExponential: y = 5 · 3ˣ
Example: From two points
An exponential function has f(0) = 6 and f(1) = 9. Write it.
- a = f(0) = 6.
- b = 9 ÷ 6 = 1.5.
Answerf(x) = 6 · 1.5ˣ
Try it yourself
Work each one on paper first, then open the answer to check.
A population of 500 doubles every year. How big is it after 4 years?
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8,000
What is the growth factor for 8% growth a year?
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1.08
What is the decay factor for a 20% loss each year?
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0.8
$2,000 grows at 10% a year, compounded yearly. How much is there after 2 years?
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$2,420
A 64 g sample halves every day. How much is left after 5 days?
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2 g
Is y = 4, 7, 10, 13 linear or exponential?
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Linear: it adds 3 each time.
Is y = 2, 6, 18, 54 linear or exponential?
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Exponential: it multiplies by 3 each time.