B. Exponential Functions · Exponential Functions

Exponential Growth and Decay: Recognizing Models and Their Restrictions

B2.2About 70 minutes

Learning goals

I can identify exponential growth and decay from a table, a graph or an equation.
I can read the initial value and the percent rate of growth or decay from y = a(1 + r)ᵗ or y = a(1 − r)ᵗ.
I can write models that use a doubling time or a half-life, such as y = a(2)^(t/d) and y = a(1/2)^(t/h).
I can explain the restrictions a real situation places on the domain and range of a model.

Words to know

exponential growth
A quantity multiplied by the same factor b > 1 in each equal time step.

exponential decay
A quantity multiplied by the same factor b with 0 < b < 1 in each equal time step.

growth rate / decay rate (r)
The percent change per step, written as a decimal. Growth: b = 1 + r. Decay: b = 1 − r.

doubling time
The time it takes a growing quantity to double.

half-life
The time it takes a decaying quantity to fall to half. Often used for radioactive substances.

restriction
A limit on the domain or range that comes from the real situation, such as t ≥ 0 or whole numbers only.

Same kind of curve, very different stories

A town grows 2% a year. A car loses 15% of its value a year. A sample of iodine-131, a radioactive form of iodine used in medicine, loses half its radioactivity about every 8 days. A pan of hot soup cools toward the temperature of the kitchen.

All four are exponential, but each one comes with its own limits: a population cannot be 3.7 people, a car's value cannot be negative, soup will not cool below the kitchen's temperature. In this lesson you will recognize growth and decay models and say exactly which values of the variables make sense.

Recognizing growth and decay

In y = a(b)ᵗ with a > 0:

Growth if b > 1: the graph rises; the table values are multiplied by more than 1 each step.
Decay if 0 < b < 1: the graph falls toward 0; the table values are multiplied by less than 1.

Percent form

If a quantity grows by r percent per step, multiply by (1 + r) each step. If it decays by r percent, multiply by (1 − r):

Growth: y = a(1 + r)ᵗ Decay: y = a(1 − r)ᵗ

Examples: 1.035 means 3.5% growth; 0.92 means 8% decay (1 − 0.92 = 0.08); 2 means 100% growth (doubling).

Left: growth by 15% per step from a = 100. Right: decay by 25% per step from a = 400. In both, a is the starting value on the vertical axis.

Doubling time and half-life

Sometimes a situation tells you how long it takes to double or to halve:

Doubling every d units of time: y = a(2)^(t/d). The exponent t/d counts "how many doublings" have happened.
Halving every h units of time (half-life h): y = a(1/2)^(t/h) or y = a(0.5)^(t/h).

For example, a bacteria culture that starts at 500 and doubles every 3 hours: N = 500(2)^(t/3). After 12 hours there have been 12/3 = 4 doublings: N = 500 × 2⁴ = 8000. After 1 hour, N = 500(2)^(1/3) ≈ 630 (a rational exponent, so a fraction of a doubling).

Worked example

Example 1: Growth or decay? What rate?

For each model, state growth or decay, the initial value and the percent rate per step. (a) P = 12 000(1.025)ᵗ (b) V = 28 000(0.85)ᵗ (c) A = 50(0.5)^(t/8) (d) N = 300(3)ᵗ

(a) b = 1.025 > 1: growth of 2.5% per year; initial value 12 000.
(b) b = 0.85 < 1: decay of 1 − 0.85 = 0.15 = 15% per year; initial value 28 000.
(c) Decay, with a half-life of 8 (time units); initial value 50.
(d) b = 3: growth; the amount triples each step, which is a 200% increase per step; initial value 300.

growth 2.5%; decay 15%; decay, half-life 8; growth 200% (tripling)

Worked example

Example 2: Write a model from a description

Write a model for each. (a) A rural municipality has 4200 residents and grows by 1.8% per year. (b) A laptop bought for $1500 loses 30% of its value each year. (c) A sample of iodine-131 has activity 80 units; its half-life is about 8 days.

(a) Growth: b = 1 + 0.018 = 1.018. P = 4200(1.018)ᵗ, t in years.
(b) Decay: b = 1 − 0.30 = 0.70. V = 1500(0.70)ᵗ, t in years.
(c) Half-life 8 days: A = 80(0.5)^(t/8), t in days.

P = 4200(1.018)ᵗ; V = 1500(0.7)ᵗ; A = 80(0.5)^(t/8)

Restrictions from the context

The function y = a(b)ˣ by itself has domain all real numbers and range y > 0. A real situation usually allows less:

Domain (time): usually t ≥ 0 (from when the model starts). A model might only be valid up to some end time, for example "from 2025 to 2035". Negative t means "before the start" and only makes sense if the same pattern really held in the past.
Discrete domains: bounce numbers, generations or full years counted as whole numbers: t = 0, 1, 2, …
Range: populations are whole numbers; values of objects cannot be negative; a cooling object stays above the room temperature and at or below its starting temperature.
Model limits: no population grows at the same percent forever (it runs out of food or space), and a ball eventually stops bouncing. Models are useful only within reasonable limits.

Soup at 85°C cooling in a 20°C kitchen. For t ≥ 0, the temperature stays in the range 20 < T ≤ 85 (shown by the shaded bar beside the axis). The model only describes the soup after it is served, so negative values of t are outside the domain.

Worked example

Example 3: State the restrictions

State a reasonable domain and range for each model. (a) Soup: T = 20 + 65(0.9)ᵗ, t in minutes after it is served. (b) Ball: h = 150(0.6)ⁿ, n = bounce number, for the first 5 bounces. (c) Town: P = 4200(1.018)ᵗ, valid for the next 10 years.

(a) Domain: t ≥ 0 (time since serving). Range: 20 < T ≤ 85. T starts at 85 and approaches 20 but never reaches it.
(b) Domain: n ∈ {0, 1, 2, 3, 4, 5} (whole bounces). Range: the matching heights {150, 90, 54, 32.4, 19.44, 11.664} cm.
(c) Domain: 0 ≤ t ≤ 10. Range: from 4200 up to 4200(1.018)¹⁰ ≈ 5020, and only whole numbers of people: about 4200 ≤ P ≤ 5020.

See each step.

Just for fun

A math teacher told her class that bacteria doubling every 20 minutes would, in theory, outweigh Earth in a few days.

A student asked, "So why hasn't it happened?" The teacher said, "Because real bacteria run out of food, and the model runs out of being true." That is a domain restriction in action.

Hands-on activity

Model hunt around the house

You need: paper, a pencil and a calculator. No experiment needed.

Find three things at home or in the news that could grow or decay by a roughly constant percent. Ideas: savings in a bank account, the value of a phone or bike over time, a sourdough starter or yeast dough rising, a mug of tea cooling, a phone battery that loses a similar fraction of charge each hour on standby (roughly).
For each one, invent reasonable numbers and write a model in the form y = a(1 ± r)ᵗ.
State whether it is growth or decay, and give a reasonable domain and range.
Say when the model would stop being realistic.

What to look for: sample: a $900 phone losing 25% a year: V = 900(0.75)ᵗ; decay; domain 0 ≤ t ≤ 6 (years you might own it); range 900(0.75)⁶ ≈ 160 ≤ V ≤ 900. The model stops being realistic once the phone is broken or no longer sold used. Each model should have a starting value, a base that matches the percent, and limits that fit the situation.

Your turn

Growth or decay, and what percent per step? y = 75(1.12)ᵗ

Growth or decay, and what percent per step? y = 75(0.88)ᵗ

Write a model: 2500 fish, decreasing 6% per year.

Write a model: 40 mg of a substance with a half-life of 5 hours.

Write a model: 120 bacteria doubling every 30 minutes (t in minutes).

Table: t = 0, 1, 2; y = 800, 680, 578. Growth or decay? Rate?

Give a reasonable domain for the number of fish in a lake modelled for the next 8 years, and explain one range restriction.

More practice: warm-up

Is y = 9(0.3)ˣ growth or decay?

What base gives 7% growth?

What base gives 40% decay?

What is the initial value of y = 650(1.4)ᵗ?

In A = 10(0.5)^(t/3), what is the half-life?

More practice: on your own

A painting is worth $5000 and gains 6% a year. Write a model and find its value after 4 years.

N = 500(2)^(t/3) models bacteria (t in hours). Find N after 6 h and after 4 h.

A cooling model is T = 22 + 70(0.8)ᵗ. Give the domain and range if t is minutes after the drink is poured.

A ball dropped from 3 m rebounds to 60% of its previous height. Write the model and the range for the first 4 bounces (n = 0 to 4).

Why should a model for the number of visitors to a new provincial park trail, P = 300(1.2)ᵗ (t in weeks), not be used for t = 52?

Challenge

A population triples every 10 years and is 2000 now. Write a model. Then find the yearly growth factor and percent (two decimal places).

The half-life of carbon-14 is about 5730 years. What fraction of the original carbon-14 remains after 17 190 years?

Two models start at 1000: A = 1000(0.9)ᵗ and B = 1000 − 100t. Compare their domains and ranges if both describe a quantity that cannot be negative.

Check your understanding

Quiz

Which model shows exponential decay?

A quantity grows by 4.5% per year. What is its growth factor?

What percent decay does y = 300(0.82)ᵗ show?

A substance has a half-life of 12 hours and starts at 64 g. Which model fits?

A drink cools by T = 21 + 60(0.88)ᵗ. What is the range for t ≥ 0?

For the number of a ball's bounces, the domain should be:

For parents and teachers

Big idea (B2.2): students identify exponential functions, including real growth and decay situations, from tables, graphs and equations, and explain the restrictions a context puts on the domain and range.

How to help: Use real-life phrases: "loses 15% a year", "doubles every 3 hours", "half-life of 8 days". Ask your teen to turn each into an equation and to say what values of t and y make sense.

Common difficulties:

Using r instead of 1 + r: 3% growth is a factor of 1.03, not 0.03 or 3.
Decay factor: 15% decay is 0.85, not 0.15.
Half-life exponent: (0.5)^(t/h), not (0.5)^(ht).
Forgetting real-world limits: whole people, values that cannot be negative, temperatures that stay above room temperature, and models that only work for a limited time.

A note on facts: iodine-131's half-life is about 8 days and carbon-14's is about 5730 years; these values are widely published. All other numbers in this lesson are made-up but realistic examples.