Growth & Decay Functions
Growth vs Decay — Equations
Sort each function into the correct column: Growth or Decay.
Identify the Growth Factor
Circle the growth or decay factor in each function.
A = 5000 × 1.06ⁿ — the growth factor is:
V = 20000 × 0.85ⁿ — the decay factor is:
P = 300 × 1.15ⁿ — the percentage growth rate is:
M = 1000 × 0.92ⁿ — the percentage decrease per step is:
Percentage to Growth Factor
Draw a line from each percentage change to the corresponding growth/decay factor.
Starting Value
Circle the starting value (initial amount) in each function.
A = 5000 × 1.06ⁿ
P = 250 × 0.9ⁿ
y = 3 × 2ˣ
Growth vs Decay — Real World
Sort each scenario: Growth or Decay?
Equation to Graph Description
Draw a line from each equation to the description of its graph behaviour.
What Does the Graph Look Like?
Circle the correct description of each function's graph.
y = 2ˣ starts at y = 1 and:
y = 100 × 0.5ⁿ starts at 100 and:
y = 3x + 5 is:
Key Features of Exponential Graphs
Circle the correct answer.
All graphs of y = aˣ (where a > 0) pass through:
For exponential decay (0 < a < 1), as x → ∞, y approaches:
The graph of y = 2ˣ has a horizontal asymptote at:
Compound Interest Steps
Put the steps for calculating compound interest in the correct order.
Compound Interest Calculations
Circle the correct answer for each compound interest problem.
$1,000 at 10% for 2 years compounded annually:
$5,000 at 6% for 3 years compounded annually:
$2,000 at 5% for 1 year compounded annually:
Simple vs Compound Interest
Circle the correct comparison.
$10,000 at 8% simple interest for 5 years earns:
$10,000 at 8% compound interest for 5 years grows to:
The difference between compound and simple interest after 5 years is:
Match the Formula
Draw a line from each description to its formula.
Car Depreciation
A car is bought for $40,000 and depreciates at 20% per year. Circle the correct answer.
Value after 1 year:
Value after 2 years:
Value after 3 years:
The depreciation function is:
Depreciation vs Straight Line
Circle the correct answer about depreciation methods.
Declining balance depreciation (20% per year) means the amount lost each year:
Straight-line depreciation (same amount each year) is modelled by:
After many years, declining balance depreciation means the asset value approaches:
Doubling Time Estimates
Use the rule of 70 (doubling time ≈ 70 ÷ rate%) to answer each question.
At 6% annual growth, an investment doubles in approximately:
At 9% annual growth, doubling takes approximately:
To double your money in 4 years, you need a rate of approximately:
At 3% inflation, prices double in approximately:
Write the Function
Write an appropriate function for each scenario.
A car worth $35,000 depreciates by 18% per year. Write a function V(n) for its value after n years.
A town's population is 12,000 and grows at 2.5% per year. Write a function P(n) for the population after n years.
A cup of coffee at 90°C cools toward room temperature (22°C). The temperature difference halves every 10 minutes. Write a function T(t) for the temperature after t minutes.
Car Depreciation Problem
Solve the following depreciation problem. Show all working.
A car worth $35,000 depreciates by 18% per year. Calculate its value after 4 years. Is it worth more or less than half the original price? After how many years will it first be worth less than $10,000?
Investment Doubling
Solve the following investment problem. Show all working.
You invest $2,000 at 6% per annum compounded annually. Use the rule of 72 to estimate how many years until the investment doubles. Then verify using the compound interest formula A = P(1 + r)ⁿ by calculating A for your estimated year.
Comparing Growth Models
Compare different growth models for the same scenario.
A small business earns $50,000 in its first year. Model A assumes revenue grows by $5,000 per year (linear). Model B assumes revenue grows by 8% per year (exponential). Calculate revenue for years 1–5 under each model. In which year does Model B first exceed Model A? Which model is more realistic for a growing business, and why?
Half-Life Problem
Apply exponential decay to radioactive half-life.
A 100 g sample of a radioactive substance has a half-life of 3 years. Write a decay function M(t). Calculate the mass remaining after 9 years. After how many years will less than 1 g remain?
Population Modelling
Model population growth with exponential functions.
Australia's population was approximately 20 million in 2004 and 26 million in 2024. Assuming exponential growth, calculate the annual growth rate. Use your model to predict the population in 2034. What assumptions does your model make?
True or False — Growth & Decay
Circle TRUE or FALSE.
Exponential growth means the same amount is added each period
A decay factor of 0.85 means a 15% decrease each period
Compound interest always gives more than simple interest for the same rate and time
An exponential function can have a negative base
Financial Growth Investigation
Investigate real-world financial growth and decay.
- 1Research the current interest rates at two different banks. Calculate how much $5,000 would grow to after 5 years at each rate. Which bank gives the better return?
- 2Track the depreciation of a family car: find its original purchase price and current value, then calculate the average annual depreciation rate.
- 3Use an online compound interest calculator to explore how changing the rate by 1% affects the result over 10, 20, and 30 years.
Growth & Decay in Nature
Find examples of growth and decay in the natural world.
- 1Research how bacteria populations grow. If one bacterium doubles every 20 minutes, how many would there be after 8 hours? Why doesn't this actually happen in practice?
- 2Look up Australia's population over the last 20 years. Plot the data points. Does it follow linear or exponential growth? Estimate the population in 2035.
- 3Research the half-life of caffeine in the human body (~5 hours). If you drink a coffee with 100 mg of caffeine at 3 pm, how much remains at bedtime (10 pm)?
Exponential Function — Key Features
Identify and explain key features of an exponential function.
For f(x) = 3 × 2ˣ, state: (a) the y-intercept, (b) whether it is growth or decay, (c) the horizontal asymptote, (d) the value when x = 4, (e) the value of x when f(x) = 96.
For f(x) = 500 × 0.8ˣ, state: (a) the initial value, (b) the decay factor, (c) the percentage decrease per step, (d) the value when x = 3, (e) when f(x) first drops below 100.
Compound Interest — Identify Variables
For each problem, circle the correct value.
A = P(1 + r)ⁿ with P = $4,000, r = 0.06, n = 5. What is A?
A = $6,050, P = $5,000, n = 2. What annual rate r was used?
$2,000 grows to $2,662.00 at 10% per year. How many years?
P = $10,000, r = 5% p.a., compounded monthly. The monthly rate is:
Financial Vocabulary — Growth and Decay
Draw a line from each financial term to its correct definition.
Linear vs Exponential vs Quadratic
Sort each function into the correct type.
Half-Life Problems
Apply exponential decay to radioactive half-life scenarios.
A radioactive isotope has a half-life of 6 hours. Starting with 800 mg: (a) Write the decay function A(t) where t is in 6-hour periods. (b) How much remains after 24 hours? After 48 hours? (c) How long until less than 10 mg remains?
Carbon-14 has a half-life of 5,730 years. An ancient artefact contains 25% of the original carbon-14. How old is the artefact?
Compound Interest — Sequence of Values
Complete the sequence of account balances.
Doubling Time
Use the rule of 70 (doubling time ≈ 70 ÷ percentage rate) to answer each question.
At 7% annual growth, money doubles in approximately:
At 2% annual growth, a population doubles in approximately:
A city growing at 3.5% per year doubles in approximately:
An investment doubling in 14 years has an approximate annual growth rate of:
Comparing Growth Models
Compare linear and exponential growth for the same starting value.
Company A grows by $500 per year (linear). Company B grows by 8% per year (exponential). Both start with $5,000 revenue. Complete a table showing revenue for years 1–5, then determine in which year Company B overtakes Company A.
Sketch or describe the graphs of both companies over 10 years. At what point does the exponential curve become dramatically different from the linear one? Why?
Decay Factor to Percentage Decrease
Draw a line from each decay factor to the correct percentage decrease per period.
Depreciation — Real Asset
Apply exponential decay to model vehicle depreciation.
A car costs $32,000 new and depreciates at 18% per year. (a) Write the depreciation function V(n). (b) Find the value after 3 years. After 7 years. (c) When does the car first become worth less than $10,000? (d) Calculate the total decrease in value over the first 5 years.
Real-World Growth Rate Identification
Circle the correct growth/decay rate for each real-world scenario.
Australia's population grows at roughly 1.6% per year. The growth factor is:
A phone loses 30% of its value each year. The decay factor is:
Inflation at 3.5% per year means prices grow by factor:
A drug clears 25% from the bloodstream per hour. After 2 hours, the fraction remaining is:
Spreadsheet Modelling
Design a spreadsheet to model compound growth or decay.
Describe how you would set up a spreadsheet to model $20,000 invested at 6.5% compound interest for 20 years. What columns would you create? What formula would go in the balance column? What would the final balance be?
Using the same spreadsheet, add a column for 'simple interest' at the same rate. At which year does compound interest produce $5,000 more than simple interest?
Which Model Fits?
Sort each description: which function type best models it?
Compound Interest Research
Explore real compound interest products available in Australia.
- 1Look up the current interest rates on Australian savings accounts. If you deposited $5,000, how much would you have after 10 years? Compare two different banks using A = P(1 + r)ⁿ.
- 2Research what a 'term deposit' is. Find a current term deposit rate and calculate the total return on a $10,000 deposit for 1, 2, and 5 years. Compare to a savings account.
- 3Interview a family member about saving or investing. Ask what interest rate their savings earn and whether it compounds monthly, quarterly, or annually. Calculate the effective annual rate.
The Power of Time in Investing
Explore the effect of time on compound growth.
Person A invests $5,000 at age 20 at 7% annual compound interest. Person B invests $10,000 at age 40 at the same rate. Both retire at age 65. Who has more money? Show all calculations.
Explain in your own words why starting to invest early is so important. Use the concept of exponential growth to justify your answer.
Growth vs Decay Examples from Research
Record examples of growth and decay you find in one week of reading news or textbooks.
| Item | Tally | Total |
|---|---|---|
Economic growth examples | ||
Population growth examples | ||
Radioactive/medical decay | ||
Depreciation examples | ||
Environmental/ecology examples |
Exponential Decay — Drug Dosage
Model drug concentration in the bloodstream using exponential decay.
A painkiller is eliminated at 20% per hour. A patient takes 400 mg at 8 am. (a) Write the function C(t) for concentration after t hours. (b) Find the concentration at noon, at 6 pm, and at midnight. (c) A second dose of 400 mg is taken at 8 pm. What is the total concentration just after the second dose? (d) Why is it important for doctors to know the half-life of a drug?
Exponential Equations — Solve by Inspection
Circle the value of x that solves each equation.
2ˣ = 32
3ˣ = 81
10ˣ = 0.001
5ˣ = 1
(1/2)ˣ = 8
Inflation and Purchasing Power
Apply exponential growth to model inflation.
Inflation averages 2.5% per year. A coffee costs $5.00 today. (a) Write the price function P(n) after n years. (b) What will the coffee cost in 10 years? In 30 years? (c) How many years until the price doubles? (Use the rule of 70.) (d) If your salary also grows at 2.5% per year from $60,000, what will it be in 10 years?
Continuous vs Discrete Growth
Draw a line from each scenario to whether it is best modelled by continuous or discrete exponential growth.
Interpret a Growth Function Graph
Circle the correct interpretation for each feature of an exponential graph.
The y-intercept of y = 3 × 2ˣ is:
The horizontal asymptote of y = 500 × 0.9ˣ is:
The graph of y = 4 × 3ˣ is steeper than y = 2 × 3ˣ because:
Logarithms and Exponential Equations in Finance
Use logarithms to solve financial equations with unknown time.
How long does it take for an investment to triple at 6% compound interest per year? Using A = P(1.06)ⁿ = 3P, divide both sides by P to get 1.06ⁿ = 3. Take log₁₀ of both sides: n × log(1.06) = log(3). Solve for n. Show all steps.
A car depreciates at 15% per year. Its initial value is $25,000. After how many years will it be worth less than $5,000? Set up the equation and solve using logarithms.
Simple vs Compound Interest Features
Sort each statement: True of Simple Interest Only, Compound Interest Only, or Both.
Continuous Compounding and Euler's Number
Explore the mathematical limit of compound interest.
When interest is compounded n times per year, the formula is A = P(1 + r/n)^(nt). Calculate the value of $1 invested at 100% interest for 1 year when compounded: (a) annually (n=1), (b) monthly (n=12), (c) daily (n=365), (d) each minute (n=525,600). What number does the result approach? This is Euler's number e ≈ 2.71828.
Superannuation — Long-Term Compound Growth
Apply exponential growth to Australian superannuation.
Australia requires employers to contribute 11% of a worker's salary to superannuation. If a person earns $60,000/year and their super earns 7% p.a. compound: (a) How much is contributed each year? ($60,000 × 0.11 = ?) (b) Assuming contributions are invested at the start of each year, use the geometric series sum to estimate the total after 40 years of work. (c) Compare this to simply saving the same amount without compounding.
Exponential vs Linear — Which Grows Faster?
Circle the correct answer about comparing growth rates.
In the long run, y = 1,000,000x (linear) vs y = 2ˣ (exponential). Which is larger for very big x?
The function y = 10x and y = 1.01ˣ. For x = 1,000, which is larger?
For small values of x (x = 1 to 5), which grows faster: y = 100x or y = 2ˣ?
Steps for Modelling Exponential Growth from Data
Put the steps in order for fitting an exponential model to data.
Logistic Growth — When Exponential Models Break Down
Explore the limits of exponential growth models.
Bacteria in a petri dish initially double every 30 minutes. After a few hours, the growth slows as nutrients run out. (a) Sketch what you would expect the population graph to look like over 24 hours. (b) Explain why an exponential model works at first but eventually fails. (c) What type of curve (S-shaped or logistic) better describes the full picture?
Types of Exponential Problems Practised
Tally the types of exponential problems you solve in a practice session.
| Item | Tally | Total |
|---|---|---|
Compound interest calculations | ||
Depreciation problems | ||
Population growth problems | ||
Half-life / radioactive decay | ||
Drug concentration problems |
Superannuation and Investment Research
Research real investment and superannuation options in Australia.
- 1Visit the MoneySmart website (moneysmart.gov.au). Use their superannuation calculator to see how a 15-year-old today might grow their superannuation by retirement at age 67. Try different contribution and return rate assumptions.
- 2Research the 'rule of 72' (another version of the rule of 70). Verify it by testing: at 8% growth, does money double in 72/8 = 9 years? Use the compound interest formula to check.
- 3Find the 10-year average return of the Australian stock market (ASX 200). Use this rate to calculate what $10,000 invested today would be worth in 30 years. Compare to keeping the money in a savings account.
Half-Life and Radioactive Decay
Apply exponential decay to radioactive substances.
Iodine-131 has a half-life of 8 days. A sample starts at 200 mg. Write an equation for the amount remaining after t days.
How much iodine-131 remains after 24 days? After 40 days?
How long until only 10 mg remains? Solve algebraically using logarithms.
Carbon-14 has a half-life of 5,730 years. If a fossil has 30% of its original C-14 remaining, estimate its age.
Growth and Decay Vocabulary
Match each term to its correct definition.
Identify Growth or Decay
Circle whether each equation represents growth or decay.
P = 500 × (1.08)^t
A = 1000 × (0.95)^t
N = 200 × (1.005)^t
V = 50,000 × (0.85)^t
Compound Interest vs Simple Interest
Compare simple and compound interest and understand the difference.
Calculate the simple interest on $5,000 at 6% p.a. for 4 years. Write the formula used.
Calculate the compound interest on $5,000 at 6% p.a. compounded annually for 4 years. Show full working.
How much more does compound interest earn than simple interest over 4 years? Explain why the difference grows over time.
Compound Interest Formula — Identify Components
Circle the correct answer about the compound interest formula A = P(1 + r)^n.
What does P represent?
For weekly compounding at 12% p.a., what is r per period?
Monthly compounding over 3 years gives n =
The term (1 + r)^n is called the
Depreciation and Reducing Balance
Apply exponential decay to asset depreciation.
A car is purchased for $35,000 and depreciates at 18% per year. Write the equation for its value V after t years.
Find the car's value after 3 years and after 7 years.
After how many years will the car's value first fall below $10,000? Solve using logarithms.
Compare this to straight-line depreciation at the same annual dollar amount. Which model better reflects reality? Explain.