Last updated October 13, 2026.

Secondary Math I · Module 2 · Linear & Exponential Functions

2.4 — Getting Down to Business

Supplement for Mathematics Vision Project (MVP) curriculum · Secondary Math I, Module 2, Section 2.4
Independently created to support students using MVP Math. Not official publisher material and not affiliated with or endorsed by the Mathematics Vision Project.

Today’s goal. Build and compare linear and exponential growth models, estimate when one overtakes another, explain why exponential growth accelerates, and write equations from tables and graphs. Review arithmetic and geometric sequences and their domains.

1. In class: A Solidify Understanding Task

Build two business-growth models

IN YOUR MVP PACKETSection 2.4 · Main task · p. 22 · Parts a–b

Write a table, equation, and graph for the two companies; use millions of dollars as the income unit.

Two companies begin with different net incomes in 2010. Net income means the money left after expenses. We will measure it in millions of dollars per year, and let t be the number of years after 2010. Thus t=0 means 2010, and t=5 means 2015.

Calcu-rama begins at $5 million and plans to add $0.5 million every year. When a quantity increases by the same amount each year, use a linear model.

Worked example 1 — The linear company

At the starting year, t=0, net income is 5 (million dollars). Each additional year adds 0.5:

t (years since 2010)	0	1	2	3	4
Calcu-rama (million $)	5	5.5	6	6.5	7

Why: 5 is the starting amount and 0.5 is the increase per year. The graph is a straight line with slope 0.5. At t=8: , or $9 million.

For yearly checkpoints, an equivalent recursive rule is , for .

Computafest begins at $2 million and aims to grow by 15% of the previous year's amount each year. A 15% increase means that the next value is 100% + 15% = 115% of the preceding one.

Worked example 2 — The exponential company

The first few yearly values are:

t	0	1	2	3	4
Computafest (million $)	2	2.30	2.645	3.04175	3.4980125

Why: each new value is the previous value multiplied by 1.15. The recursive form at whole-number years is .

The increase in the first year is million. In the third year it is million. Unlike Calcu-rama's fixed $0.5 million, the amount added increases over time.

Constant difference versus constant percentage

A linear model adds the same amount each year: . An exponential growth model multiplies by the same factor each year: , with . Never treat a 15% annual increase as adding a fixed 0.15 million dollars.

Compare the companies over time

IN YOUR MVP PACKETMain task · pp. 22–23 · Parts b–d

Use the two models to compare projected income, decide when one overtakes the other, and discuss long-run growth.

At first, Calcu-rama has higher projected income. Computafest starts much smaller, but as its income grows, each 15% increase becomes larger in actual dollars.

Worked example 3 — Compare the same years

Year	t	Calcu-rama	Computafest
2010	0	5.00	2.00
2020	10	10.00	8.09
2022	12	11.00	10.70
2023	13	11.50	12.31
2030	20	15.00	32.73

In 2022 Calcu-rama is still ahead. At the 2023 annual checkpoint, Computafest has overtaken it. At t=20 the exponential model predicts more than twice the linear model's income.

Linear and exponential projections from the MVP business problem (in millions of dollars per year). The plotted curves agree at about t=12.29; the highlighted points indicate selected whole-year projections.

Finding when the models match

We want to know when . We can write the equation:

This equation contains t both as a regular number and in an exponent, so the algebra methods learned so far do not isolate t easily. Use a table or a graph, then estimate between the last year one company leads and the first year the other leads.

Worked example 4 — Bracket the crossing

At : Calcu-rama is 11.00 and Computafest is about 10.70, so Computafest is still behind.
At : Calcu-rama is 11.50 and Computafest is about 12.31, so Computafest is ahead.
The models therefore meet between t=12 and t=13 if we treat both as smooth functions of elapsed time.
Using a graphing calculator or numerical estimate gives years after 2010. This is an approximation from the chosen models, not a guaranteed business outcome.

If only annual checkpoints are modeled, there is no whole-number year when the two projected values are exactly equal. The first whole-number year Computafest comes out ahead is 2023.

Will they meet again in future years? By 2023, the exponential company's next year's increase is already larger than Calcu-rama's fixed $0.5 million increase: million. Future increases under the exponential model continue growing, while Calcu-rama always adds $0.5 million. Thus Computafest stays ahead for later years in these models.

Packet clarification — Main task, Part c (investment choice)

The exercise asks which company you would choose to invest in. Projected net income alone is not enough to decide which investment is better. The exercise can be answered as a comparison of the two growth models; a real investment decision would also depend on stock price, risks, whether the forecasts come true, and many other factors.

Choose what the model tells you between years

IN YOUR MVP PACKETMain task · p. 23 · Part e

Explain when discrete yearly forecasts or a continuous-time approximation is useful.

Two models can use the same formula but have different domains (permitted inputs).

Model choice	Domain and meaning	Graph
Yearly snapshots	t=0,1,2,3,...; the model provides a value only at each annual checkpoint	Isolated points
Smooth projection	Every real time t≥0; the model estimates values between annual checkpoints	Connected line or curve

A business's actual yearly net income is normally determined over a reporting period, not smoothly earned at exactly the predicted rate. That doesn't prevent a smooth curve from being a useful approximation for planning. A discrete-year model avoids suggesting we know the precise income at every instant.

Clarifying MVP — Main task, Part e

Discrete does not mean “outputs are whole numbers.” Net income can take decimal values in millions of dollars, even if we calculate it only at whole-number years. The discrete or continuous domain depends on which input times our model permits. A smooth formula is not evidence that actual business income changes smoothly at every instant.

2. READY — Complete arithmetic and geometric sequences

Study the worked examples first, then try the problems yourself. An answer key at the bottom of this section is provided to check your work.

IN YOUR MVP PACKETReady · p. 24 · Problems 1–3

Fill three missing terms between a given first and fifth term for both an arithmetic and a geometric sequence.

Arithmetic: four equal additions

From the first to the fifth term there are four steps, not five. If the first term is a and the fifth is z, each arithmetic step adds

Worked example 5 — Arithmetic terms between two endpoints

The first term is 8 and the fifth is 28. There are four steps, so .

Term	1st	2nd	3rd	4th	5th
Value	8	13	18	23	28

Check: each step adds 5, and four additions of 5 give the total change of 20.

Geometric: four equal multiplications

For a geometric sequence with common ratio r, the fifth term is the first term multiplied by r four times:

We usually try a simple positive ratio first. But because a negative number multiplied by itself four times is positive, a negative common ratio can also work when four steps connect endpoints of the same sign.

Worked example 6 — Two correct geometric sequences

First term 3; fifth term 48:

Both and satisfy this.

Ratio	1st	2nd	3rd	4th	5th
+2	3	6	12	24	48
−2	3	−6	12	−24	48

Both use a constant multiplication factor and reach the same fifth term. For purely geometric sequences on integer indices, negative ratios are valid. For the usual real-valued exponential growth functions, the exponential base is positive.

MVP clarification — Section 2.4, Ready, Problems 1–3

The worked example uses a positive common ratio, but Problems 1–3 do not restrict the sign of the ratio. Both positive and negative common ratios give valid geometric sequences for these problems.

Module 1, Section 1.10 (“Geometric Meanies”) explicitly teaches students to consider whether geometric sequences can have more than one solution. That applies here, too.

For example, in Problem 1, both 3, 6, 12, 24, 48 and 3, −6, 12, −24, 48 are valid geometric sequences. Their common ratios are 2 and −2, respectively.

Both sequences satisfy the given first and fifth terms. The packet does not say that only positive ratios are allowed.

When the endpoints are negative, the pattern works the same way. For example, a sequence beginning at −80 and ending at −5 has positive ratio or negative ratio , since .

Check your work — Ready 1–3 answer key

Arithmetic: $14.25,25.5,36.75$. Geometric: $6,12,24$; ratio $-2$ also gives $-6,12,-24$.
Arithmetic: $-4690,-3130,-1570$. Geometric: $-1250,-250,-50$; ratio $-\frac15$ also gives $1250,-250,50$.
Arithmetic: $-9.1875,-6.375,-3.5625$. Geometric: $-6,-3,-1.5$; ratio $-\frac12$ also gives $6,-3,1.5$.

3. SET — Distinguish sequences from continuous-domain models

Study the worked examples first, then try the problems yourself. An answer key at the bottom of this section is provided to check your work.

IN YOUR MVP PACKETSet · pp. 24–25 · Problems 4–11

Recognize domain symbols, compare recursive and explicit rules, and connect arithmetic sequences with linear models and geometric sequences with exponential models.

Read the domain notation

The domain is the set of allowed inputs. Sets in the homework are written using symbols such as , , , , and .

Symbol	Meaning at this level	Examples
	Real numbers, including decimals and fractions	−1, 0, 0.4, √2, 3
	Rational numbers (can be written as a fraction)	−1/2, 0, 2/3, 7
	Integers (no fractional parts)	−2, −1, 0, 1, 2
	Whole numbers	0, 1, 2, 3, ...
	Natural numbers; often 1,2,3,..., but some textbooks include 0	1, 2, 3, ...

For example, means “all nonnegative real inputs,” including 0, 0.2, 1.5, and 10. A function can be continuous on that interval, but naming a domain by itself does not prove the function is continuous.

Worked example 7 — A yearly sequence versus a smooth function

Compare the two expressions:

Both use a linear rule. The first is a discrete arithmetic sequence with separate allowed inputs. The second is a continuous linear function defined for all nonnegative real times. The rule is the same; the domain is different.

Recursive and explicit descriptions

A recursive sequence describes how the next term follows the previous. An explicit rule computes a value directly from the input:

Pattern	Recursive (whole-number index n)	Explicit
Arithmetic
Geometric

In the packet, formulas with and a given starting value are intended as recursive descriptions over whole-number steps. In contrast, an explicit formula such as can naturally be used to define a continuous exponential curve when the real-number domain is chosen. An equation alone may be restricted to either domain; the choice must be stated or understood from context.

Worked example 8 — Identify the function family from an equation

Compare and .

For , each increase of 1 in t adds 2. For , each increase of 1 multiplies by 2. That makes p linear and q exponential. Both could be sampled only at integers or modeled on a continuous real domain; this doesn't change their growth pattern.

MVP clarification — Section 2.4, Set, Problem 8

D is the expected answer in this textbook. Arithmetic and geometric sequences are conventionally indexed by natural numbers, often 1, 2, 3, ...

However, C is also mathematically valid. A sequence can begin at index 0, giving the domain {0, 1, 2, 3, ...}. Both starting conventions are standard; some textbooks also include 0 among the natural numbers.

For Set Problem 9, the attributes shared by arithmetic and geometric sequences include having separated input values, having some consistent feature (difference or ratio), and admitting recursive rules. The specific domain is often a subset of the integers; not every sequence must start at the same index or extend forever.

Check your work — Set 4–11 answer key

A: nonnegative real inputs.
D: $\{x\mid x\in\mathbb N\}$.
B: $g(x)=5\cdot4^x$.
C: $h(t)=3t-5$.
D for a sequence starting at 1; C is also appropriate when indexing starts at 0. Both use separate whole-number indices.
B, C, F, G under the packet’s positive-integer indexing: discrete; natural-number domain; constant difference or ratio; a recursive rule.
Arithmetic sequences follow linear rules at separate term indices. A continuous linear function includes all real inputs in its stated interval.
Geometric sequences follow exponential rules at separate term indices. A continuous exponential function includes fractional inputs too.

4. GO — Write equations from tables and graphs

Study the worked examples first, then try the problems yourself. An answer key at the bottom of this section is provided to check your work.

IN YOUR MVP PACKETGo · p. 26 · Problems 12–21

Recover explicit equations from four numerical tables and six graphs, including a bonus challenge.

From a table: check equal differences and equal factors

Use the x-values as well as the outputs. When x increases by 1 each row:

Same addition	Same multiplication
Linear:	Exponential:

GO worked Problem A — Linear rule from a table

x	2	3	4	5
f(x)	1	−3	−7	−11

Each increase of 1 in x decreases f by 4, so . Insert into :

Answer: . Check with x=5: .

GO worked Problem B — Exponential rule with a negative input

x	−2	−1	0	1
f(x)	32	16	8	4

Each output is half the previous output, so . At x=0, , giving

Check: . A negative x is allowed in an exponential rule and does not make the output negative.

From a graph: use actual coordinates

For a straight line, read two precise grid-intersection points, calculate , then find b from one point. For an exponential curve, read the y-intercept (if visible) to get a, then compare values at two consecutive x-values to find the factor b.

GO worked Problem C — A graph's two linear points

A graphed line goes through and .

Use : , so . The equation is . Check both points by substitution.

GO worked Problem D — An exponential curve

An increasing exponential curve crosses the y-axis at 3 and passes through . In , the y-intercept gives ; at x=1 we have , so .

Check another point: when x=2 the model predicts 12, and when x=−1 it predicts 1.5.

Go Problems 12–15: The tables include negative outputs and negative input values. A valid exponential model may have a negative starting multiplier a while retaining a positive base b. For example, is always negative but doubles in magnitude with each 1-unit increase of x.

Packet clarification — Go, Problem 21 (optional bonus)

The bonus graph crosses the x-axis and curves downward; it is not a basic function of the form , because those functions never cross the x-axis when and . A vertically shifted exponential could produce that shape, but that is an extension beyond the simplest equation-writing method used in Problems 12–20. Do not infer that any downward-curving graph must be .

Quick check before accepting an equation

Initial value: Does substituting x=0 (when available) reproduce the y-intercept?
Pattern: Does the line add the same output difference per unit x, or does the exponential multiply by a fixed factor?
Test: Does your equation correctly reproduce at least two other listed points?
Graph: Is the plotted direction and shape consistent with the sign of the slope or the exponential base?

Check your work — Go 12–21 answer key

For the unlabeled grids in 16–21, these equations assume one square represents one unit on each axis. A different scale changes the numerical equations.

$f(x)=-7x+10$.
$f(x)=2\cdot5^x$.
$f(x)=-6\cdot2^x$.
$f(x)=\left(\frac13\right)^x$.
$y=-\frac13x+2$.
$y=4^x$.
$y=-x+1$.
$y=\frac12\cdot2^x$.
$y=-4x-4$.
One compatible shifted exponential is $y=2-2^x$.

Additional Learning Resources

Resource & direct link	What it teaches	Scope / What to skip
Khan Academy — Linear vs. exponential growth from data	Identify linear and exponential patterns from tables and build growth models; particularly useful for main task parts a–d and Go tables.	Close match. Focus on deciding which pattern fits the inputs and outputs.
Paul's Online Math Notes — Exponential Functions	Explains exponential form and interprets its graph, with worked examples.	Some extra material. Use the opening definition and introductory graphs; skip the longer discussion of the special constant e and advanced graph behavior.
Professor Leonard — Exponential Growth and Decay (College Algebra 66)	Optional extended lecture about percent-change models.	Significantly broader. Focus on basic percent growth and decay; skip half-life and Newton's law of cooling. Specific timestamps could not be verified.
OpenStax — 4.2 Modeling with Linear Functions	Additional worked explanations of converting an initial amount and constant growth rate into a linear formula.	Some extra material. Read the opening model-building examples; skip advanced fitting/estimation discussions.

Teacher's Review of the Original MVP Packet

The central business-comparison task has a sound mathematical purpose, and the teacher's manual explicitly invites either discrete-year or smooth modeling assumptions. The following are the issues that warrant clarification.

Set, Problem 9 — “Always” needs a domain convention

The prompt treats a sequence domain as though it must be the entire set of natural numbers. But finite sequences, zero-indexed sequences, and other integer-indexed sequences are legitimate. A more precise statement is that an arithmetic or geometric sequence has an integer-indexed domain (often consecutive nonnegative or positive integers), a constant difference or ratio, and an appropriate recursive description.

Go, Problems 16–21 — Reading graphs and extending the model

Students must infer grid scales where numerical labels are not shown, and the optional Problem 21 is a curve crossing the x-axis, unlike a basic unshifted exponential. The worksheet should state “one grid square represents one unit” where intended and clearly label Problem 21 as requiring a different, shifted exponential model. The bonus designation appropriately signals extra challenge, but it does not specify the new function form.

Curriculum Coverage

MVP assignment	Required skills	Where taught
Main a–b (p. 22)	Construct and compare linear/exponential equations, tables, and graphs	In class; worked examples 1–3; original comparison graph
Main c–d (p. 23)	Interpret forecasts; estimate crossover using table/graph; understand long-run growth	In class; worked example 4 and packet clarification
Main e (p. 23)	Choose discrete-year versus smooth real-domain models	In class
Ready 1–3 (p. 24)	Four equal arithmetic steps; geometric ratios and sign ambiguity	Section 2; worked examples 5–6 and red clarification
Set 4–5 (p. 24)	Real, rational, integer, whole, and natural-number domain sets	Section 3, notation table
Set 6–11 (p. 25)	Recursive/explicit rules; domains and arithmetic/geometric families	Section 3; worked examples 7–8 and red clarification
Go 12–15 (p. 26)	Recover linear and exponential rules from tables	Section 4; worked examples A–B
Go 16–21 (p. 26)	Recover equations from graphs; distinguish the optional extension	Section 4; worked examples C–D and bonus clarification

Worked examples are original, not copies of the packet answers. Packet page numbers above refer to printed pages, not the PDF viewer's page numbers.

Last updated October 13, 2026.