Teaching unit · Algebra 1 (commonly taken Grades 8 to 10)

Exponential functions: growth, decay and geometric sequences

Telling linear and exponential change apart, then constructing both kinds of function from input-output pairs

About four lessons of 45 to 60 minutes

Start here · hook

Adding the same amount vs. multiplying by the same amount

Two savings plans both start at $100. Plan A adds $20 every month. Plan B multiplies by 1.2 every month (a 20% increase). After one month they are equal at $120. After two years, Plan A is worth $580, while Plan B is $100 x $1.2^{24}$ , about $7,949.68: nearly 80 times its starting value and more than 13 times Plan A. Adding the same amount every step (linear growth) and multiplying by the same amount every step (exponential growth) look similar at first and diverge dramatically.

Everything in this unit comes from that one distinction: linear functions grow by equal DIFFERENCES over equal intervals, exponential functions grow by equal FACTORS. Once you can tell which one a situation is, you can construct the actual function, and predict its value at any point, exactly.

A bacteria culture doubling every hourmultiplying by 2 each hour is exponential growth, not adding a fixed number of cells
A car losing 15% of its value every yearexponential DECAY, multiplying by 0.85 each year, never a fixed dollar drop
Compound interest on a savings accountthe classic real-world exponential growth model, F-LE.A.1.c's 'constant percent rate'
A geometric sequence like 3, 6, 12, 24, ...each term is the previous term times a fixed common ratio, the discrete version of an exponential function

Learning objective

What students will be able to do

Students will distinguish linear from exponential tables using equal differences vs. equal factors, construct linear functions f(x) = mx + c and exponential functions f(x) = a . $b^{x}$ from input-output pairs, find terms of arithmetic and geometric sequences, and solve constant-percent growth and decay word problems exactly.

Success criteria

I can decide whether a table is linear or exponential by checking for a constant difference or a constant factor between consecutive y-values.
I can find m and c and write f(x) = mx + c from two input-output pairs.
I can find a and b and write f(x) = a . $b^{x}$ from two input-output pairs, including pairs that do not show f(0).
I can find any term of an arithmetic or geometric sequence from its first term and common difference or ratio.
I can compute the exact value of a quantity after several years of constant-percent growth or decay.

Curriculum anchor

Standards this unit teaches

HSF-LE.A.1.aCommon Core (US)
Equal differences vs. equal factors
Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
HSF-LE.A.1.cCommon Core (US)
Constant percent growth and decay
Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
HSF-LE.A.2Common Core (US)
Construct linear and exponential functions
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Exponential function: a function of the form f(x) = a . $b^{x}$ , where a is the starting value and b is the constant growth/decay factor
Growth factor: the number you multiply by each step; greater than 1 for growth, between 0 and 1 for decay
Arithmetic sequence: a list of numbers where each term is the previous term plus a fixed common difference, the discrete version of a linear function
Geometric sequence: a list of numbers where each term is the previous term multiplied by a fixed common ratio
Common ratio: the fixed multiplying factor between consecutive terms of a geometric sequence
Exponential decay: repeated multiplication by a factor between 0 and 1, so the quantity shrinks toward zero without ever reaching it

Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. Linear vs. exponential: equal differences vs. equal factors

Concrete

Given a table of x and y values with x increasing by steady steps (0, 1, 2, 3, ...), there is a quick, certain test for which kind of relationship it is: check the y-values. If the DIFFERENCE between consecutive y-values stays the same, the table is linear. If the FACTOR (the ratio) between consecutive y-values stays the same instead, the table is exponential.

This is not a coincidence, it follows from the definitions. y = mx + b changes by exactly m every time x increases by 1 (an equal DIFFERENCE), because m(x+1) + b - (mx + b) = m every time. y = a . $b^{x}$ changes by a FACTOR of b every time x increases by 1, because a . $b^{x+1}$ / (a . $b^{x}$ ) = b every time, regardless of x.

Worked example

Two tables both start at x = 0. Table A: 5, 8, 11, 14. Table B: 3, 6, 12, 24. Which is linear and which is exponential?

Table A differences: 8-5=3, 11-8=3, 14-11=3. The difference is constant (3), so Table A is linear.
Table A factors (to check they are NOT constant): 8/5=1.6, 11/8=1.375. Not constant, confirming it is not exponential.
Table B differences: 6-3=3, 12-6=6, 24-12=12. NOT constant, so Table B is not linear.
Table B factors: 6/3=2, 12/6=2, 24/12=2. The factor is constant (2), so Table B is exponential.

Answer: Table A is linear (constant difference of 3); Table B is exponential (constant factor of 2).

Check for understanding, ask

Could a table have BOTH a constant difference and a constant factor? Try to build one.
If a table's y-values are all the same number, is it linear, exponential, or both?
Why does checking just the FIRST pair of differences or factors never prove a table is linear or exponential?

2. Constructing linear and exponential functions from input-output pairs

Pictorial

Once a relationship is classified, two input-output pairs determine its function. For a linear function, find the slope m = (y2 - y1)/(x2 - x1), then substitute either pair into c = y - mx. For an exponential function with consecutive inputs, the factor is b = y2/y1; then recover a from a = y1/b^x1. The especially simple pair f(0), f(1) gives a = f(0) and b = f(1)/f(0) directly.

For example, the pairs (1, 5) and (3, 11) are linear: m = (11 - 5)/(3 - 1) = 3, then c = 5 - 3(1) = 2, so f(x) = 3x + 2. The exponential pairs f(1) = 12 and f(2) = 36 have b = 36/12 = 3 and a = 12/ $3^{1}$ = 4, so f(x) = 4 . $3^{x}$ . An ARITHMETIC sequence is the discrete linear case, term n = a1 + (n - 1)d; a GEOMETRIC sequence is the discrete exponential case, term n = a1 . $r^{n-1}$ .

f(x) = 5 .

3^{x}

, from the worked example. Each step to the right multiplies the height by 3, an exponential curve, not a straight line.

Worked example

A table shows f(0) = 5 and f(1) = 15. Write f(x) = a . $b^{x}$ , then find f(4).

a = f(0) = 5.
b = f(1) / f(0) = 15 / 5 = 3.
f(x) = 5 . $3^{x}$ .
f(4) = 5 . $3^{4}$ = 5 x 81 = 405.

Answer: f(x) = 5 . $3^{x}$ ; f(4) = 405.

Check for understanding, ask

Why does f(0) alone tell you the value of a, with no other information needed?
If instead f(1) = 5 and f(2) = 15, would b still equal 3? Explain.
Given the linear pairs (1, 5) and (3, 11), how do the slope and intercept calculations produce f(x) = 3x + 2?
How does a1 + (n - 1)d for an arithmetic sequence relate to f(x) = mx + c for a linear function?
How does a1 . $r^{n-1}$ for a geometric sequence relate to f(x) = a . $b^{x}$ for an exponential function?

3. Constant-percent growth and decay

Abstract

A real-world quantity described as growing or decaying by a constant PERCENT rate is a disguised exponential function: convert the percent to a growth factor first (100% + the growth percent, or 100% - the decay percent, written as a decimal or fraction), then that factor plays the role of b, multiplied once per time period.

Growing by 50% each year means multiplying by 1.5 (or 3/2) every year: the new amount is 150% of the old one. Decaying by 25% each year means multiplying by 0.75 (or 3/4) every year: only 75% of the amount remains. After t years, the exact value is (starting amount) x $(factor)^{t}$ , the same exponential structure as section 2, just applied to a percent-rate story problem.

Worked example

A car is worth $6,400 and depreciates (loses value) by 25% each year. Find its exact value after 2 years.

Losing 25% each year means 75% remains: the yearly factor is 3/4.
After 2 years: $6,400 x $(3/4)^{2}$ .
$(3/4)^{2}$ = 9/16.
$6,400 x 9/16 = $57,600 / 16 = $3,600.

Each year multiplies the value by 3/4, not subtracting a fixed dollar amount, so the drop from year 1 to year 2 ($1,200) is smaller than the drop from year 0 to year 1 ($1,600).

Answer: $3,600.

Check for understanding, ask

Why is the yearly factor for 25% decay equal to 3/4, not 1/4?
What yearly factor would a 10% GROWTH rate give?
Why does the dollar amount lost get smaller every year, even though the PERCENT lost stays the same?

Watch for

Common misconceptions and how to address them

MisconceptionA quantity that grows or decays by a percentage each period changes by the same AMOUNT each time, just like a linear function.

Why it happens: Percent language ('loses 25% every year') sounds like a fixed, repeatable action, easy to confuse with a fixed dollar amount.

How to address it: A constant PERCENT rate is a constant FACTOR, not a constant amount. As the total shrinks (or grows), the same percentage represents a smaller (or larger) actual amount every period, exactly what makes it exponential rather than linear.

MisconceptionDecaying by 25% each year means multiplying by 0.25.

Why it happens: Students take the percent given in the problem and convert it directly to a decimal, skipping the 'what remains' step.

How to address it: Losing 25% leaves 75% REMAINING. The multiplying factor is 100% - 25% = 75% = 0.75, not the 25% that was lost. Always ask 'what fraction is left?', not 'what fraction is gone?'.

MisconceptionChecking just the first two y-values of a table is enough to prove it is linear or exponential.

Why it happens: The first difference or factor is the easiest one to compute, so it feels like sufficient evidence.

How to address it: A single pair only rules things IN, never proves the pattern holds for the WHOLE table. Check at least two more consecutive pairs (three total) before concluding a table is linear or exponential.

MisconceptionThe nth term of a geometric sequence is a1 . $r^{n}$ .

Why it happens: Students count the multiplications the same way they count the term number, forgetting the first term itself needed zero multiplications.

How to address it: Term 1 is a1 . $r^{0}$ = a1 (no multiplying yet). Term 2 is a1 . $r^{1}$ . In general, term n is a1 . $r^{n-1}$ , ONE FEWER than the term number.

Do it together

Guided practice (with answers)

1. A table shows y-values 12, 12, 12, 12 as x increases by 1 each time. Is it linear, exponential, both, or neither?
Answer: Both: the difference is constant (0) AND the factor is constant (1), a special constant function that fits both definitions.
2. A table shows f(0) = 4 and f(1) = 20. Write f(x) = a . $b^{x}$ .
Answer: f(x) = 4 . $5^{x}$ , because a = f(0) = 4 and b = f(1)/f(0) = 20/4 = 5.
3. A linear function contains the input-output pairs (1, 5) and (3, 11). Construct f(x) = mx + c.
Answer: f(x) = 3x + 2, because m = (11 - 5)/(3 - 1) = 3 and c = 5 - 3(1) = 2.
4. An arithmetic sequence begins 5, 8, 11, ... Find the 10th term.
Answer: 32, because the common difference is 3 and term 10 = 5 + (10 - 1) x 3 = 32.
5. A geometric sequence begins 2, 10, 50, ... Find the 5th term.
Answer: 1250, because the common ratio is 5, and term 5 = 2 x $5^{4}$ = 2 x 625 = 1250.
6. A population of 400 grows by 10% each year. Find the exact population after 2 years.
Answer: 484, because the yearly factor is 11/10, and 400 x $(11/10)^{2}$ = 400 x 121/100 = 484.
7. A sample of 300 grams decays by 50% each hour. Find the exact mass after 3 hours.
Answer: 37.5 grams, because the hourly factor is 1/2, and 300 x $(1/2)^{3}$ = 300/8 = 37.5.

On their own

Independent practice worksheets

Practise telling linear and exponential tables apart, constructing linear and exponential functions and geometric sequences, and solving growth/decay word problems, with computed, never-wrong answer keys and exact values.

Algebra 1: linear vs. exponential tablesthe core distinction from section 1, freshly generated each time Algebra 1: constructing linear & exponential functionspractise section 2, recovering f(x) = mx + c or f(x) = a . b^x from pairs, plus arithmetic and geometric sequence terms Algebra 1: exponential growth & decayconstant-percent word problems from section 3, every answer a whole number Grade 8: functions, slope and linear equationsrevisit the linear side of the comparison

Reach every student

Differentiation

Support

Start distinguishing linear from exponential with tables where the exponential factor is a whole number (x2, x3), before introducing decay (unit fraction) factors.
Provide a small reference table of powers of common bases (2, 3, 5, 10) while evaluating $b^{x}$ .
Use only whole-number growth factors (never percent decay) for the first construct-the-function examples.
Give growth/decay word problems with the yearly factor already computed, so converting a percent to a fraction is not a barrier to practising the exponent step.

Extension

Construct an exponential function from non-consecutive points, for example f(2) = 36 and f(5) = 972: solve $b^{3}$ = 972/36 = 27, so b = 3, then recover a = 36/ $3^{2}$ = 4.
Compare a linear and an exponential model that start at the same value and ask when the exponential overtakes the linear one, using a table or the functionGraph figure as a starting point.
Introduce a repeated-percent problem with a NEGATIVE overall change requested (e.g. 'how many years until the value drops below half?') as an extension into inequalities.
Research a real compound-interest or population-growth statistic and model it with an exponential function, then check the model's prediction against the real reported figure.

Check it stuck

Assessment: exit ticket

A short exit ticket sampling the linear/exponential distinction, constructing both kinds of function, and a growth/decay word problem.

1. A table shows y-values 7, 10, 13, 16. Linear or exponential? Justify.
Answer: Linear, because the difference between consecutive y-values is constant (3), not the factor.
2. A linear function contains the pairs (2, 9) and (5, 21). Construct f(x) = mx + c.
Answer: f(x) = 4x + 1, because m = (21 - 9)/(5 - 2) = 4 and c = 9 - 4(2) = 1.
3. A table shows f(0) = 6 and f(1) = 18. Write f(x) = a . $b^{x}$ and find f(3).
Answer: f(x) = 6 . $3^{x}$ ; f(3) = 6 x 27 = 162, because a = 6 and b = 18/6 = 3.
4. An investment of $800 grows by 25% each year. Find its exact value after 2 years.
Answer: $1,250, because the yearly factor is 5/4, and 800 x $(5/4)^{2}$ = 800 x 25/16 = 1250.

For the teacher

Teacher notes and timings

Rough timing across four lessons: Lesson 1 equal differences vs. equal factors (section 1), Lesson 2 constructing f(x) = mx + c and f(x) = a . $b^{x}$ plus geometric sequences (section 2), Lesson 3 constant-percent growth and decay (section 3), Lesson 4 mixed practice plus the exit ticket.
Prior knowledge: Grade 8's linear-functions unit (grade8LinearFunctions, linked above) develops y = mx + b in full. Section 2 gives the focused two-pair construction refresh needed to cover the linear as well as exponential part of HSF-LE.A.2.
Scope note versus AU's Financial Mathematics (buildFinancialSecondary, content_ausecondarymath.ts): that unit computes simple interest, a single discount, or a single tax addition, always ONE step. This unit's whole point is COMPOUNDING over multiple periods, the genuinely exponential case that unit never reaches.
Language to repeat: a constant PERCENT rate is a constant FACTOR (multiply), never a constant AMOUNT (add); 'what fraction remains' (for decay) is the number to raise to a power, not 'what fraction is lost'.
Curriculum note: HSF-LE.A.1.a is the proof/justification skill (section 1's core, 'equal differences vs. equal factors'); HSF-LE.A.2 is the construct-the-function skill (section 2 covers both linear and exponential functions and geometric sequences); HSF-LE.A.1.c is the applied percent-rate skill (section 3). Verified live at thecorestandards.org/Math/Content/HSF/LE/A/1/a/, /c/ and /A/2/ on 2026-07-14.

All teaching units Make a worksheet