Exponential growth and decay (Algebra II) · lesson

Writing and using a · b^t

💡 In a · b^t, a is the starting amount and b is what you multiply by each step.

To find a value, do the power first, then multiply by a.

To write a model, find a and b. Growing by r percent means b = 1 + r. Shrinking by r percent means b = 1 − r. Write r as a decimal: 6% is 0.06.

🧠 Let's work it out

A town of 800 grows 6% per year. Write the rule for P(t).

Start value: a = 800.
Growth factor: b = 1 + 0.06 = 1.06.
So P(t) = 800 · 1.06^t.

What do you think comes next?

exponential function: a function where the variable is in the exponent, like 3 · 2^x
evaluate: find the value by plugging in a number

🔗 Khan Academy: Exponential models · Khan Academy (free) ↗

Growth or decay? Read the base

💡 With a > 0, the function grows when b > 1 and decays when 0 < b < 1; the starting value doesn't decide it.

The base b is the growth factor. If b is more than 1, each step makes the amount bigger. If b is between 0 and 1, each step makes it smaller.

The percent change is the distance from 1. A base of 0.97 keeps 97%, so it loses 3% each step. A base of 1.25 gains 25%.

🧠 Let's work it out

A car's value is V(t) = 9,000 · (0.97)^t. What percent does it lose per year?

0.97 is between 0 and 1, so the value decays.
1 − 0.97 = 0.03.
0.03 is 3% lost per year.

What do you think comes next?

growth factor: the number you multiply by each step
decay: shrinking by the same percent each step

Half-life and doubling time

💡 Count how many half-lives or doublings fit in the time, then halve or double that many times.

A half-life is the time it takes for half of something to be gone. A doubling time is the time it takes to double.

Divide the total time by the half-life or doubling time. That tells you how many steps to take. Make the units match first: 1 hour is 60 minutes.

🧠 Let's work it out

A sample of 96 g has a half-life of 5 days. How much is left after 20 days?

20 ÷ 5 = 4 half-lives.
Halve it 4 times: 96 → 48 → 24 → 12 → 6.

What do you think comes next?

half-life: the time for an amount to drop to half
doubling time: the time for an amount to double

Compound interest

💡 Compounded n times a year, A = P(1 + r/n)^(nt); compounded continuously, A = Pe^(rt).

P is the starting amount, r is the yearly rate as a decimal, and t is the number of years. Monthly means n = 12. Quarterly means n = 4.

Continuous compounding uses the number e, about 2.718. Use the e^x key on a calculator.

Round money to the nearest cent only at the very end.

🧠 Let's work it out

Put $1,000 in at 5% compounded quarterly for 2 years. How much do you have?

Rate per quarter: r/n = 0.05 ÷ 4 = 0.0125.
Number of quarters: nt = 4 × 2 = 8.
So A = 1000 × 1.0125⁸.
That is about 1104.486, so $1,104.49.

What do you think comes next?

compound interest: interest that also earns interest
e: a special number, about 2.718, used for continuous growth

Finding a rate from the factor

💡 Rewrite the factor for the time step you want, then read the percent as the distance from 1.

If a factor covers one year, its square root covers half a year. For example, 1.69^t = (1.3²)^t = 1.3^(2t). Each half-year multiplies by 1.3, which is 30% growth.

If you know two values, set start · b^t = end. Divide, then take a root to find b.

🧠 Let's work it out

A town grew from 100 to 196 people in 2 years at the same yearly rate. What is the yearly growth rate?

Two years of growth: 100 · b² = 196.
Divide by 100: b² = 1.96.
Take the root: b = √1.96 = 1.4.
1.4 means 40% growth per year.

What do you think comes next?

rate: the percent change for each step