Teacher Guide: Exponential Function Word Problems

Solve real exponential word problems with one 5-step method: compound interest, depreciation, viral spread, half-life and continuous growth.

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Class quiz

10 questions on Exponential Functions. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Apply a 5-step method to set up and solve exponential word problems
Choose the correct model: compound interest, continuous growth, half-life or a general $a \cdot b^t$
Find the growth or decay factor from a percentage rate or from a table of values
Decide from a table whether a pattern is linear or exponential
Interpret $a$ and $b$ in context and judge whether an answer is reasonable

Prerequisites

• Introduction to Exponential Functions: the form $f(x) = a \cdot b^x$
• Exponential Growth: growth factor $1 + r$ and compound interest
• Exponential Decay: decay factor $1 - r$ and depreciation
• Half-Life Problems: the model $N_0 \left(\frac{1}{2}\right)^{t/T}$
• Continuous Growth and Decay: the model $P \cdot e^{rt}$ with $e \approx 2.71828$

Discussion Starters

1. A bank offers 6% compounded yearly, another 5.9% compounded monthly. How would you decide which is better?
2. Why does a car lose more money in its first year than in its fifth year, even though the percentage is the same?
3. A post doubles its shares every hour. Can that continue for a whole week? What would stop it?
4. What words in a problem tell you to use a factor instead of adding the same amount each time?

Common Misconceptions

A 15% decrease per year means multiplying by 0.15

Remediation: Ask: "If you lose 15 of every 100 dollars, how many dollars stay?" Write $100 - 15 = 85$ , so the factor is $0.85$ . Then check the result is still a large part of the start after one year.

The exponent is always the number of years

Remediation: Make students count periods aloud: "monthly for 10 years is 120 months", "half-life of 6 hours over 18 hours is 3 half-lives". Write the exponent as periods, not years.

Any increasing table is exponential

Remediation: Give two increasing tables, one with constant differences and one with constant ratios. Students compute both differences and ratios and classify each one.

Differentiation Ideas

For Struggling Students:

• Give the 5-step organizer with the formula already chosen in step 3
• Start with whole-number factors (doubling, tripling, halving) before percentages
• Let students list the first 3 or 4 periods by hand before using the formula

For On-Level Students:

• Mix compound interest, depreciation and half-life problems in one set
• Compare yearly, quarterly and monthly compounding for the same deposit
• Classify tables as linear or exponential and write the model

For Advanced Students:

• Find the growth factor from two data points, like $b^2 = 1.44$
• Compare an offer with continuous compounding to one with monthly compounding
• Write their own word problem and a wrong solution for a classmate to correct

Standards Alignment

HSF-LE.A.1 (CCSS.MATH.CONTENT.HSF.LE.A.1)
Distinguish between situations that can be modeled with linear functions and with exponential functions
HSF-LE.A.2 (CCSS.MATH.CONTENT.HSF.LE.A.2)
Construct linear and exponential functions given a graph, a description of a relationship, or two input-output pairs (include reading these from a table)
HSF-LE.B.5 (CCSS.MATH.CONTENT.HSF.LE.B.5)
Interpret the parameters in a linear or exponential function in terms of a context
HSA-CED.A.1 (CCSS.MATH.CONTENT.HSA.CED.A.1)
Create equations in one variable and use them to solve problems, including equations arising from exponential functions

Lesson Resources

visualCompound Interest Calculator
Change the compounding period and see how monthly beats yearly over 10 years
activityWhich Model? Sorting Cards
Students sort 12 short scenarios into linear, exponential growth, exponential decay and continuous
worksheet5-Step Word Problem Organizer
A template with one box per step: read, identify, model, compute, check

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

An exponential word problem describes a quantity that changes by the same factor in every equal time period. Your job is to turn the story into a model of the form

y = a \cdot b^{t}

where

a

is the initial value and

b

is the growth factor (

b > 1

) or decay factor (

0 < b < 1

The 5-step method works for every problem in this lesson:

1.Read: find the starting amount, the rate or factor, the time and the units.
2.Identify: is it growth or decay? Find the factor $b$ (for a rate $r$ : $b = 1 + r$ or $b = 1 - r$ ).
3.Write the model: choose the right formula and put in the numbers.
4.Compute: evaluate with a calculator and round as the problem asks.
5.Check: is the answer reasonable (bigger or smaller than the start?) and in the right units?

Formula recap (from the earlier lessons):

Compound interest, $n$ times per year: $A = P\left(1 + \frac{r}{n}\right)^{nt}$
Continuous growth or decay: $A = P \cdot e^{rt}$
Half-life $T$ : $N(t) = N_0 \left(\frac{1}{2}\right)^{t/T}$

Worked Examples

You deposit 5000 dollars in a savings account that pays 6% annual interest, compounded monthly. How much is in the account after 10 years?

Read: list the quantities

$P = 5000$ dollars, $r = 0.06$ , $n = 12$ (monthly), $t = 10$ years → Start, rate, periods per year and time found

Identify the factor per month

$1 + \frac{r}{n} = 1 + \frac{0.06}{12} = 1.005$ → Growth, factor $1.005$ per month

Write the model

$A = 5000 \cdot 1.005^{12 \cdot 10} = 5000 \cdot 1.005^{120}$ → $A = 5000 \cdot 1.005^{120}$

Compute

$1.005^{120} \approx 1.819397$ , so $A \approx 5000 \cdot 1.819397 \approx 9096.98$ → $A \approx 9096.98$

Check

Yearly compounding gives $5000 \cdot 1.06^{10} \approx 8954.24$ . Monthly should give a little more, and it does. → Reasonable, in dollars

Answer: After 10 years the account holds about 9096.98 dollars.

Common Mistakes

Using $r$ and $t$ instead of $\frac{r}{n}$ and $nt$ in compound interest

Why it's wrong: With monthly compounding the bank pays $\frac{6\%}{12} = 0.5\%$ each month, 120 times in 10 years. Writing $5000 \cdot 1.06^{120}$ or $5000 \cdot 1.005^{10}$ mixes up the period.

Correct: Rate per period $\frac{r}{n}$ , number of periods $nt$ : $A = 5000\left(1 + \frac{0.06}{12}\right)^{12 \cdot 10}$ .

Using the rate as the factor, for example $b = 0.15$ for a 15% loss

Why it's wrong: A 15% loss means 85% of the value stays. $24000 \cdot 0.15^5$ is almost zero, which no car does in 5 years.

Correct: Decay: $b = 1 - r = 0.85$ . Growth: $b = 1 + r$ . The factor is what is left (or what you have) after one period.

Using $t$ instead of $\frac{t}{T}$ when the period is not 1

Why it's wrong: "Triples every 2 hours" or "half-life of 6 hours" means the factor applies once per 2 or 6 hours, not every hour. $40 \cdot 3^{10}$ is far too big.

Correct: Count the periods first: exponent $= \frac{\text{time}}{\text{length of one period}}$ , for example $3^{10/2} = 3^5$ .

Skipping the reasonableness check

Why it's wrong: A decay answer larger than the start, or a growth answer smaller than the start, always signals a wrong factor or a wrong exponent.

Correct: Step 5 every time: growth ends above $a$ , decay ends between $0$ and $a$ , and the units match the question.

Why It Matters

Exponential models are the language of money, medicine, technology and nature. Knowing how to set them up yourself means you can check claims instead of just believing them:

Savings and loans: the difference between yearly and monthly compounding is real money over 10 or 20 years
Buying a car or a phone: depreciation tells you what it will be worth when you sell it
Health: half-life tells a doctor how much medicine is still in your body after a few hours
Social media: viral posts grow by a factor in each time period, which is why they explode so fast
Science: populations, radioactive materials and cooling objects all follow exponential laws

This lesson brings together everything from the exponential functions unit. The 5-step method is the same every time; only the formula in step 3 changes.

Real World Applications

Saving for University

Banks compound interest yearly, quarterly, monthly or daily. The formula $A = P\left(1 + \frac{r}{n}\right)^{nt}$ lets you compare offers before you choose.

Example:

1000 dollars at 4% compounded quarterly for 3 years: $1000 \cdot 1.01^{12} \approx 1126.83$ dollars.

1Try It Yourself

You put 4000 dollars into an account paying 6% annual interest, compounded quarterly, for 2 years.

How much money is in the account after 2 years (to the nearest cent)?

Step 1: Write the mathematical expression

Use $A = P\left(1 + \frac{r}{n}\right)^{nt}$ with $P = 4000$ , $r = 0.06$ , $n = 4$ , $t = 2$ :

Bacteria in the Lab

Microbiologists model growth that happens all the time, not in steps, with continuous growth $A = P \cdot e^{rt}$.

Example:

A culture of 1000 cells growing continuously at 2% per hour for 10 hours: $1000 \cdot e^{0.2} \approx 1000 \cdot 1.2214 \approx 1221$ cells.

2Try It Yourself

A culture starts with 800 cells and grows continuously at a rate of 3% per hour. Use $e^{0.3} \approx 1.3499$ .

About how many cells are there after 10 hours (to the nearest whole cell)?

Step 1: Write the mathematical expression

Use $A = P \cdot e^{rt}$ with $P = 800$ , $r = 0.03$ , $t = 10$ :

Caffeine in Your Body

Doctors and pharmacists use half-life to decide how often a medicine is taken. Caffeine has a half-life of about 5 hours in an adult.

Example:

A 100 mg coffee at 8:00 still leaves $100 \cdot \left(\frac{1}{2}\right)^{1} = 50$ mg at 13:00.

3Try It Yourself

An energy drink contains 160 mg of caffeine. Assume a half-life of 5 hours.

How much caffeine is left after 10 hours?

Step 1: Write the mathematical expression

Use $N(t) = N_0 \left(\frac{1}{2}\right)^{t/T}$ with $N_0 = 160$ , $T = 5$ , $t = 10$ :

Key Takeaways

1Every exponential word problem fits $y = a \cdot b^{t}$ : $a$ is the starting value, $b$ the factor per period
2Use the 5 steps: read, identify growth or decay and the factor, write the model, compute, check
3A rate $r$ gives the factor $b = 1 + r$ for growth and $b = 1 - r$ for decay
4Compound interest: $A = P\left(1 + \frac{r}{n}\right)^{nt}$ ; continuous: $A = P \cdot e^{rt}$ ; half-life: $N_0 \left(\frac{1}{2}\right)^{t/T}$
5When the factor applies once every $T$ units of time, the exponent is $\frac{t}{T}$
6In a table, constant differences mean linear, constant ratios mean exponential

Frequently Asked Questions

How do I know whether a problem is exponential and not linear?

Look for words like "percent per year", "doubles", "triples", "halves" or "half-life": the change is a factor. If the story says "increases by 50 each year", the change is an amount, so it is linear. In a table, test the ratios of consecutive values.

When do I use $e^{rt}$ instead of $(1 + r)^t$ ?

Use

e^{rt}

only when the problem says the growth or decay is continuous (or compounded continuously). If it says yearly, monthly, quarterly or gives no compounding at all, use

(1 + r)^t

\left(1 + \frac{r}{n}\right)^{nt}

My calculator has no $e$ key. What do I do?

Use the value given in the problem, like

e^{0.4} \approx 1.4918

, or type

2.71828

for

e

. The calculator in this lesson has

e

, so there you can type e^0.4 directly. With 5 decimals of

e

your answer will round to the same cent or whole number in the problems of this lesson.

Why does the 5th step matter if the computation is correct?

Most wrong answers come from the setup, not the arithmetic: a factor of

0.15

instead of

0.85

, or

t

instead of

\frac{t}{2}

. Asking "should this be bigger or smaller than the start?" catches those errors in a few seconds.

Glossary

Exponential model: A formula $y = a \cdot b^{t}$ that describes a quantity changing by the same factor $b$ in every equal time period
Growth factor / decay factor: The number $b$ you multiply by each period: $b = 1 + r$ for growth, $b = 1 - r$ for decay
Compound interest: Interest added to the balance $n$ times per year, so later interest is also earned on earlier interest
Depreciation: The loss of value of an object (car, phone, machine) over time, often a fixed percentage per year
Half-life: The time $T$ it takes for a decaying quantity to fall to half of its value
Continuous growth: Growth that happens at every moment, modeled with $A = P \cdot e^{rt}$

Formula Card

General Exponential Model

y = a \cdot b^{t}

$a$ = initial value, $b$ = factor per period ($b = 1 + r$ growth, $b = 1 - r$ decay), $t$ = number of periods.

Compound Interest

A = P\left(1 + \frac{r}{n}\right)^{nt}

$P$ = starting amount, $r$ = annual rate as a decimal, $n$ = compounding periods per year, $t$ = years.

Continuous Growth or Decay

A = P \cdot e^{rt}

$r > 0$ for growth, $r < 0$ for decay, $e \approx 2.71828$.

Half-Life

N(t) = N_0 \left(\frac{1}{2}\right)^{t/T}

$N_0$ = starting amount, $T$ = half-life, $t$ = time passed (same unit as $T$).

Teacher Guide: Exponential Function Word Problems

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Saving for University

Bacteria in the Lab

Caffeine in Your Body

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

General Exponential Model

Compound Interest

Continuous Growth or Decay

Half-Life

More in This Topic

Introduction to Exponential Functions

Exponential Growth

Graphing Exponential Functions

Exponential Decay