Exponential Growth and Decay –

Exponential growth and decay is a Higher tier topic that models situations where a quantity increases or decreases by a constant percentage over equal time periods. Unlike linear change (which adds a fixed amount), exponential change multiplies by a fixed factor. This produces the characteristic J-shaped growth curve or the gradually flattening decay curve. Real-world examples include population growth, radioactive decay, bacterial cultures and depreciation. At GCSE, you need to recognise exponential models, use the formula y = ab^x, and interpret graphs. This guide covers all of these skills.

What Is Exponential Growth and Decay?

Exponential growth occurs when a quantity multiplies by a factor greater than 1 in each time period. The quantity increases faster and faster over time.

Exponential decay occurs when a quantity multiplies by a factor between 0 and 1 in each time period. The quantity decreases, approaching zero but never quite reaching it.

Key Formulas

Where:

• a = the initial amount (when x = 0, y = a)
• b = the growth or decay factor per time period
• x = the number of time periods
• If b > 1, it is growth. If 0 < b < 1, it is decay.

For a percentage increase of r%: b = 1 + r/100.

For a percentage decrease of r%: b = 1 - r/100.

Half-life

The half-life is the time taken for a quantity to reduce to half its value. It is constant for exponential decay. After n half-lives, the fraction remaining is:

Step-by-Step Method

1. Identify the initial amount (a) from the question.
2. Determine the growth or decay factor (b). For a 5% annual increase, b = 1.05. For a 20% annual decrease, b = 0.80.
3. Identify the number of time periods (x).
4. Substitute into y = ab^x and calculate.
5. Interpret the answer in context. State units and whether the quantity has grown or decayed.

Recognising exponential behaviour

• Exponential growth: the curve rises steeply, curving upward.
• Exponential decay: the curve falls steeply at first, then levels off, approaching but never reaching zero (an asymptote).
• In a table, check whether the ratio of consecutive values is constant (not the difference).

Worked Example 1 — Exponential Growth

Question: A colony of bacteria starts with 500 bacteria and triples every hour. How many bacteria are there after 4 hours?

Working:

a = 500, b = 3, x = 4

y = 500 x 3^4 = 500 x 81 = 40,500

Answer: After 4 hours, there are 40,500 bacteria.

Worked Example 2 — Exponential Decay

Question: A radioactive substance has a mass of 800 g and a half-life of 6 hours. Find the mass remaining after 24 hours.

Working:

Step 1: Number of half-lives = 24 / 6 = 4

Step 2: Fraction remaining = (1/2)^4 = 1/16

Step 3: Mass remaining = 800 x 1/16 = 50 g

Answer: After 24 hours, 50 g remains.

Worked Example 3 — Exam Style

Question: The value of a car when new is £18,000. It depreciates by 15% each year. Write a formula for the value V after t years. Find the value after 3 years.

Working:

Decay factor b = 1 - 15/100 = 0.85

Formula: V = 18000 x 0.85^t

After 3 years: V = 18000 x 0.85^3 = 18000 x 0.614125 = £11,054.25

Answer: V = 18000 x 0.85^t. After 3 years, the car is worth £11,054.25.

Common Mistakes

• Confusing exponential with linear. Exponential growth multiplies by the same factor. Linear growth adds the same amount. If a population doubles each year, it is exponential, not linear.
• Using b > 1 for decay. If something is decreasing, b must be between 0 and 1. A 20% decrease means b = 0.80, not 1.20.
• Forgetting the initial amount. The formula is y = ab^x, not y = b^x. The initial amount a must be included.
• Confusing half-life with halving the rate. The half-life is a time period. After one half-life, the amount halves. After two half-lives, it is a quarter of the original (not zero).
• Assuming exponential decay reaches zero. The curve approaches zero but mathematically never reaches it.

Exam Tips

• Write the formula y = ab^x and clearly state the values of a and b. This earns method marks.
• If asked to sketch the graph, show the curve starting at the initial value on the y-axis, rising steeply for growth or falling and levelling off for decay.
• For half-life questions, count the number of half-lives first, then use (1/2)^n.
• If given a table of values and asked whether growth is exponential, check whether consecutive values have a constant ratio.
• Round money answers to 2 decimal places (nearest penny) unless told otherwise.

Practice Questions

Q1 (Higher): A population of 2,000 increases by 10% per year. Find the population after 5 years.

Q2 (Higher): A sample has a mass of 640 g and a half-life of 3 hours. What mass remains after 12 hours?

Q3 (Higher): The number of downloads of an app is modelled by y = 150 x 1.25^x, where x is the number of weeks. How many downloads are there in week 6?

Practise exponential growth and decay with step-by-step solutions free on GCSEMathsAI.

Related Topics

• Compound Interest and Depreciation
• Growth and Decay
• Other Graphs: Cubic, Reciprocal, Exponential
• Reverse Compound Interest

Summary

• Exponential growth: y = ab^x with b > 1. The quantity multiplies by the same factor each period.
• Exponential decay: y = ab^x with 0 < b < 1. The quantity shrinks by the same factor each period.
• a is the initial value, b is the multiplier, x is the number of time periods.
• For a percentage increase of r%, use b = 1 + r/100. For a decrease, b = 1 - r/100.
• Half-life is the time for a quantity to halve. After n half-lives, the fraction remaining is (1/2)^n.
• Exponential curves have a characteristic shape: steep growth or decay that never crosses the x-axis.
• Check for exponential behaviour by seeing if consecutive values have a constant ratio.