Sheet № 235 · Higher only · AQA · Edexcel · OCR
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 flat
§Key definitions
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.
Question:
A colony of bacteria starts with 500 bacteria and triples every hour. How many bacteria are there after 4 hours?
Answer:
After 4 hours, there are 40,500 bacteria.
Q1 (Higher):
A population of 2,000 increases by 10% per year. Find the population after 5 years.
§Formulas to memorise
y = a x b^x
Fraction remaining = (1/2)^n
a — = the initial amount (when x = 0, y = a)
b — = the growth or decay factor per time period
x — = the number of time periods
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.
Identify the initial amount — (a) from the question.
Determine the growth or decay factor — (b). For a 5% annual increase, b = 1.05. For a 20% annual decrease, b = 0.80.
Identify the number of time periods — (x).
Worked example
A colony of bacteria starts with 500 bacteria and triples every hour. How many bacteria are there after 4 hours?
Working:
⚠ 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.