Percent Growth
& Decay

Learn to recognize when a quantity grows or decays by a constant percent rate per unit interval, and model it with exponential functions.

Algebra 2

What Is a Constant Percent Rate?

A quantity changes by a constant percent rate when the same percentage of the current value is added or removed in every equal time interval. This is fundamentally different from adding or subtracting a fixed amount.

Growth

The quantity increases by the same percent each period: population, investments, inflation.

Decay

The quantity decreases by the same percent each period: depreciation, radioactive decay, cooling.

Exponential vs. Linear Change

Real-World Examples

Savings Account

A bank pays $5\%$ interest per year. Each year, $5\%$ of the current balance is added.

Car Depreciation

A car loses $15\%$ of its value every year. After 5 years, it retains only about $44\%$ of its original value.

Bacteria Colony

A colony doubles every hour: that is a $100\%$ growth rate per hour.

The Exponential Model

The general exponential growth/decay function is:

$ a $

Initial value: the quantity at $ t = 0 $.

$ b $

Growth factor: the multiplier applied each period. $ b = 1 + r $.

$ r $

Percent rate (as a decimal). Positive for growth, negative for decay.

Understanding the Growth Factor $ b $

1

Growth: Rate $ r > 0 $, so $ b = 1 + r > 1 $. Example: $8\%$ growth means $ b = 1.08 $.

2

Decay: Rate $ r < 0 $, so $ 0 < b = 1 + r < 1 $. Example: $8\%$ decay means $ b = 0.92 $.

3

No change: Rate $ r = 0 $, so $ b = 1 $. The value stays constant.

Growth vs. Decay: Key Distinction

	Exponential Growth	Exponential Decay
Rate $ r $	$ r > 0 $	$ r < 0 $
Base $ b $	$ b > 1 $	$ 0 < b < 1 $
Behavior	Increases, curves up	Decreases, curves down
As $ t \to \infty $	$ f(t) \to \infty $	$ f(t) \to 0 $
Example	Population $+3\%$/yr	Medicine $-20\%$/hr

How to Identify Percent Growth or Decay

1

Read the context. Is a quantity changing over equal time intervals?

2

Check the language. Look for phrases like "increases by X%," "depreciates by X%," or "grows at a rate of X% per year."

3

If given a table, compute consecutive ratios $ \frac{f(t+1)}{f(t)} $. If they are constant, the change is exponential.

4

Write the model. Identify $ a $ (initial value) and compute $ b = 1 + r $. Then $ f(t) = a \cdot b^t $.

The Ratio Test for Tables

If you have a table of values, divide each value by the previous one. A constant ratio confirms exponential behavior.

Year $ t $	Value $ f(t) $	Ratio
0	500	—
1	400	$ \frac{400}{500} = 0.80 $
2	320	$ \frac{320}{400} = 0.80 $
3	256	$ \frac{256}{320} = 0.80 $

Example 1: Population Growth

A town of 12,000 people grows by $4\%$ per year. Write a model and find the population after 6 years.

1

Identify: $ a = 12{,}000 $, $ r = 0.04 $.

2

Growth factor: $ b = 1 + 0.04 = 1.04 $.

3

Model: $ P(t) = 12{,}000 \cdot 1.04^t $.

4

Evaluate: $ P(6) = 12{,}000 \cdot 1.04^6 \approx 12{,}000 \cdot 1.2653 \approx 15{,}184 $.

Example 2: Radioactive Decay

A 200-gram sample of a substance decays by $10\%$ per hour. Write a model and find the mass after 5 hours.

1

Identify: $ a = 200 $, $ r = -0.10 $.

2

Growth factor: $ b = 1 + (-0.10) = 0.90 $.

3

Model: $ M(t) = 200 \cdot 0.90^t $.

4

Evaluate: $ M(5) = 200 \cdot 0.90^5 \approx 200 \cdot 0.59049 \approx 118.1 $ g.

Example 3: Building a Model from a Table

Year	0	1	2	3	4
Value ($)	20,000	23,000	26,450	30,418	34,981

1

Ratio test: $ \frac{23{,}000}{20{,}000} = 1.15 $, $ \frac{26{,}450}{23{,}000} = 1.15 $. Constant!

2

Growth factor: $ b = 1.15 $, so $ r = 0.15 $ which is $15\%$ growth.

3

Model: $ V(t) = 20{,}000 \cdot 1.15^t $.

Graphing Growth and Decay

Writing the Model with $ r $ Explicit

The exponential model can be written to show the rate $r$ directly:

Common Mistakes to Avoid

✕ Using $ r $ as the base

Writing $ f(t) = 500 \cdot 0.05^t $ for a $5\%$ growth. The base is $ 1 + r = 1.05 $, not $ r = 0.05 $.

✕ Confusing linear and exponential

Adding a fixed dollar amount (like $50/year) is linear, not exponential. Exponential requires a percent of the current value.

✕ Forgetting to convert %

Using $ b = 1 + 15 $ instead of $ b = 1 + 0.15 $. Always convert the percent to a decimal first.

Summary Cheat Sheet

Feature	Formula / Rule
General model	$ f(t) = a \cdot b^t $
$ a $	Initial value: $ f(0) $
$ b $ (growth)	$ b = 1 + r $, where $ b > 1 $ and $ r > 0 $
$ b $ (decay)	$ b = 1 + r $, where $ 0 < b < 1 $ and $ r < 0 $
Table test	Constant ratio $ \frac{f(t+1)}{f(t)} = b $
Asymptote	$ y = 0 $ (horizontal)

Question 1 of 5

A car worth $25,000 depreciates by $12\%$ per year. Which function models its value after $ t $ years?

Depreciation of $12\%$ means the car retains $ 1 - 0.12 = 0.88 $ of its value each year, so $ b = 0.88 $ and the model is $ V(t) = 25{,}000 \cdot 0.88^t $.

Question 2 of 5

A colony of bacteria triples every hour. If there are initially 200 bacteria, which model represents this situation?

"Triples" means the quantity is multiplied by 3 each hour, so $ b = 3 $. The growth rate is $ r = 3 - 1 = 2 = 200\% $. The model is $ N(t) = 200 \cdot 3^t $.

Question 3 of 5

The table shows a savings account balance. What is the annual growth rate?

Year	0	1	2	3
Balance ($)	1,000	1,060	1,123.60	1,191.02

Compute the ratio: $ \frac{1{,}060}{1{,}000} = 1.06 $. Since $ b = 1.06 $, the rate is $ r = 0.06 = 6\% $.

Question 4 of 5

Which scenario represents exponential decay?

Losing $25\%$ of the current potency each hour is a constant percent decrease: exponential decay with $ b = 0.75 $. Adding 50 items or reading 30 pages is linear (constant amount). Growing by $7\%$ is exponential growth, not decay.

Question 5 of 5

A house is purchased for $300,000 and appreciates at $3\%$ per year. What is its value after 10 years, to the nearest dollar?

Model: $ V(t) = 300{,}000 \cdot (1.03)^t $. For $t=10$, $ V(10) = 300{,}000 \cdot (1.03)^{10} \approx 300{,}000 \cdot 1.34392 \approx \$403{,}175 $. A linear increase of $3\%$ would add $0.03 \times 300{,}000 = \$9{,}000$ each year, totaling $\$390{,}000$ after 10 years (Option B), which underestimates the compounding effect.

	Exponential Growth	Exponential Decay
Rate \( r \)	\( r > 0 \)	\( r < 0 \)
Base \( b \)	\( b > 1 \)	\( 0 < b < 1 \)
Behavior	Increases, curves up	Decreases, curves down
As \( t \to \infty \)	\( f(t) \to \infty \)	\( f(t) \to 0 \)
Example	Population \(+3\%\)/yr	Medicine \(-20\%\)/hr

Feature	Formula / Rule
General model	\( f(t) = a \cdot b^t \)
\( a \)	Initial value: \( f(0) \)
\( b \) (growth)	\( b = 1 + r \), where \( b > 1 \) and \( r > 0 \)
\( b \) (decay)	\( b = 1 + r \), where \( 0 < b < 1 \) and \( r < 0 \)
Table test	Constant ratio \( \frac{f(t+1)}{f(t)} = b \)
Asymptote	\( y = 0 \) (horizontal)

Percent Growth
& Decay

What Is a Constant Percent Rate?

Exponential vs. Linear Change

Real-World Examples

The Exponential Model

Understanding the Growth Factor \( b \)

Growth vs. Decay: Key Distinction

How to Identify Percent Growth or Decay

The Ratio Test for Tables

Example 1: Population Growth

Example 2: Radioactive Decay

Example 3: Building a Model from a Table

Graphing Growth and Decay

Writing the Model with \( r \) Explicit

Common Mistakes to Avoid

Summary Cheat Sheet

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Year \( t \)	Value \( f(t) \)	Ratio
0	500	—
1	400	\( \frac{400}{500} = 0.80 \)
2	320	\( \frac{320}{400} = 0.80 \)
3	256	\( \frac{256}{320} = 0.80 \)

Percent Growth& Decay

What Is a Constant Percent Rate?

Exponential vs. Linear Change

Real-World Examples

The Exponential Model

Understanding the Growth Factor \( b \)

Growth vs. Decay: Key Distinction

How to Identify Percent Growth or Decay

The Ratio Test for Tables

Example 1: Population Growth

Example 2: Radioactive Decay

Example 3: Building a Model from a Table

Graphing Growth and Decay

Writing the Model with \( r \) Explicit

Common Mistakes to Avoid

Summary Cheat Sheet

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Percent Growth
& Decay