How to Solve Exponential Growth and Decay Functions?

Exponential growth and decay describe situations where a quantity increases or decreases by a constant percentage each time period. Unlike linear change (which adds the same amount each period), exponential change multiplies by the same factor, leading to rapid growth or rapid decline. Population growth, radioactive decay, compound interest, and the spread of viruses are all classic examples.

What Are Exponential Growth and Decay?

In exponential growth, a quantity increases by a fixed percent each period — the amount added each time gets larger and larger. In exponential decay, a quantity decreases by a fixed percent each period — the amount removed gets smaller and smaller over time.

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The Exponential Growth and Decay Formula

\(\color{blue}{A = P(1 \pm r)}\)^t

Where:

• A = Final amount after time t
• P = Initial (starting) amount (principal)
• r = Rate of change per period (as a decimal; divide percent by 100)
• t = Number of time periods
• Use + r for growth; use − r for decay

Identifying Growth vs. Decay

Exponential Growth: \(\color{blue}{A = P(1 + r)}\)^t

The base \(\color{blue}{(1 + r)}\) is greater than 1. The value of A increases as t increases.

• P = $1,000, \(\color{blue}{r = 5}\)% = 0.05, \(\color{blue}{t = 3}\): \(\color{blue}{A = 1000(1.05)^{3} \approx $1,157.63}\).

Exponential Decay: \(\color{blue}{A = P(1 – r)}\)^t

The base \(\color{blue}{(1 – r)}\) is between 0 and 1. The value of A decreases as t increases.

• P = $2,000, \(\color{blue}{r = 10}\)% = 0.10, \(\color{blue}{t = 4}\): \(\color{blue}{A = 2000(0.90)^{4} \approx $1,312.20}\).

Step-by-Step Summary

1. Identify whether the quantity is growing (\(\color{blue}{\text{ use } +r}\)) or decaying (\(\color{blue}{\text{ use } -r}\)).
2. Convert the rate to a decimal.
3. Substitute P, r, and t into the formula \(\color{blue}{A = P(1 \pm r)^{t}}\).
4. Use a calculator or exponent rules to evaluate \(\color{blue}{(1 \pm r)^{t}}\).
5. Multiply by P to find the final amount A.

Watch: Graphing Exponential Growth & Decay

Khan Academy explains the shape and behavior of exponential growth and decay graphs:

Exponential Growth and Decay – Worked Examples

Example 1 (Growth): A savings account has $1,000 at 5% annual growth. What is the balance after 3 years?

\(\color{blue}{A = 1000(1 + 0.05)^{3} = 1000(1.05)^{3} = 1000 \times 1.157625 \approx $1,157.63}\).

Example 2 (Decay): A car purchased for $2,000 loses 10% of its value each year. What is its value after 4 years?

\(\color{blue}{A = 2000(1 – 0.10)^{4} = 2000(0.90)^{4} = 2000 \times 0.6561 \approx $1,312.20}\).

Example 3 (Growth): A town of 500 people grows at 8% per year. What is the population after 5 years?

\(\color{blue}{A = 500(1.08)^{5} \approx 500 \times 1.46933 \approx 734.66}\) (approximately 735 people).

Example 4 (Decay): A radioactive sample of 3,000 grams decays at 15% per year. How much remains after 3 years?

\(\color{blue}{A = 3000(1 – 0.15)^{3} = 3000(0.85)^{3} = 3000 \times 0.614125 \approx $1,842.37 g}\).

More Practice: Exponential Growth and Decay Word Problems

Khan Academy solves a variety of exponential growth and decay word problems:

Exercises for Exponential Growth and Decay

Use \(\color{blue}{A = P(1 \pm r)}\)^t to find the final amount. Round to the nearest cent.

1. Growth: P = $1,500, \(\color{blue}{r = 6}\)%, \(\color{blue}{t = 4}\) years.
2. Decay: P = $3,000, \(\color{blue}{r = 15}\)%, \(\color{blue}{t = 3}\) years.
3. Growth: P = $800, \(\color{blue}{r = 10}\)%, \(\color{blue}{t = 5}\) years.
4. Decay: P = $5,000, \(\color{blue}{r = 20}\)%, \(\color{blue}{t = 2}\) years.
5. Growth: P = $250, \(\color{blue}{r = 4}\)%, \(\color{blue}{t = 10}\) years.

Answers

1. \(\color{blue}{A \approx $1,893.72}\)
2. \(\color{blue}{A \approx $1,842.37}\)
3. \(\color{blue}{A \approx $1,288.41}\)
4. \(\color{blue}{A = $3,200.00}\)
5. \(\color{blue}{A \approx $370.06}\)

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Free Exponential Growth and Decay Worksheet

Ready to practice on your own? Download our free Exponential Growth and Decay worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again, steady practice is the surest way to master Exponential Growth and Decay before a quiz or test.

Download Exponential Growth Worksheet

Frequently Asked Questions

What is the formula for exponential growth and decay?

The general form is y = a·bˣ, where a is the starting amount and b is the multiplier per period. With a percent rate r, growth is y = a(1 + r)ᵗ and decay is y = a(1 − r)ᵗ. For example, something that loses 20% each year keeps 80% of its value, so its model is y = a(0.8)ᵗ.

What is the difference between exponential growth and decay?

Both multiply by the same factor each period, but growth uses a factor greater than 1 and decay uses a factor between 0 and 1. Growth, like y = 100(2)ˣ, rises faster and faster. Decay, like y = 80(½)ˣ, drops quickly at first and then levels off toward zero without ever reaching it.

What is k in exponential growth and decay?

In the continuous model A = A₀eᵏᵗ, k is the continuous growth or decay rate. A positive k means growth and a negative k means decay. If k = 0.03, the amount grows continuously at 3% per unit of time. You can find k from data by solving for it with natural logarithms, for example k = ln(A/A₀) ÷ t.

How do you solve exponential growth and decay word problems?

Find the starting amount, the rate and the number of periods, then build the model. Suppose 2,000 bacteria grow 5% per hour for 3 hours. The multiplier is 1.05, so the count is 2,000 × 1.05³ = 2,000 × 1.157625 ≈ 2,315. For decay, subtract the rate from 1 instead of adding it.

Is exponential growth and decay on the SAT?

Yes. SAT Math includes exponential functions, and questions often ask you to match a situation to a model, interpret a growth factor, or tell exponential change from linear change. Practice spotting phrases like increases by 6% each year, which points to a factor of 1.06, versus increases by 6 each year, which is linear.

How is exponential growth and decay used in the real world?

It models anything that changes by a steady percentage: compound interest, population growth, radioactive half-life, medicine leaving the body and the falling value of a car. For instance, $1,000 earning 4% interest a year becomes 1,000 × 1.04² = $1,081.60 after two years, a bit more than simple interest would give.

Related Topics

• Simple Interest
• Percent of Increase and Decrease
• Percent Problems
• Approximating Irrational Numbers
• The Distributive Property