Last updated October 13, 2026.

Secondary Math I · Module 2 · Linear & Exponential Functions

2.3 — Linear, Exponential or Neither?

Supplement for Mathematics Vision Project (MVP) curriculum · Secondary Math I, Module 2, Section 2.3
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. Recognize linear and exponential patterns from graphs, equations, tables, sequences, and stories; explain the evidence for a classification; compare changes correctly; and fill missing geometric-sequence terms and calculate slope.

1. In class: A Practice Understanding Task

Identify the pattern of change

IN YOUR MVP PACKETSection 2.3 · Main task · pp. 13–18 · Problems 1–16

Identify each type of function and give two pieces of mathematical evidence. Begin with how the output changes as the input changes.

In Sections 2.1–2.2 you worked with arithmetic sequences, geometric sequences, and their related functions. Today's task is to recognize those patterns regardless of how they are presented.

Key meanings

Linear: equal changes in the input produce equal differences in the output. For input steps of size 1, the output adds the same number each step.

Exponential: equal changes in the input produce equal multiplication factors in the output. For input steps of size 1, the output multiplies by the same positive factor each step in the standard real-valued exponential model.

Neither: the relationship does not follow either pattern. "Not linear" does not automatically mean "exponential."

Look for	Linear	Exponential
Pattern	Same amount added or subtracted	Same factor multiplied
Equation		, with and
Graph	Points follow a straight line	Typical increasing or decreasing exponential curve
What stays constant?	Slope (change in y divided by change in x)	Multiplier over equal steps in x

These forms describe the familiar nonconstant cases. A constant function is linear with slope zero, though the equation also gives the same constant outputs. This is one reason to use the pattern, not just an isolated appearance, when classifying.

Worked example 1 — An equal-difference table

Suppose the input rises by 2 at every step:

	1	3	5	7
	11	17	23	29

The y-values change by whenever x changes by . Therefore the slope is , and the rule is linear. Since at , , so . The equation is .

Two reasons: equal input steps give equal output differences; and the data agree with an equation of the form .

Worked example 2 — An equal-factor table

	0	1	2	3
	5	15	45	135

For each increase of 1 in x, the output is multiplied by 3: . The successive differences, , are not equal.

Thus is exponential. Two reasons: the repeated multiplication factor is 3; and the output is multiplied by 3 for each one-unit increase in the input, rather than adding a fixed amount.

Always check the input spacing first

Suppose the x-values are . Equal steps now mean an increase of 2, not 1. Linear outputs need equal differences across those two-unit steps. Exponential outputs need equal multiplication factors across the same steps.

Worked example 3 — A table with unequal x-steps

	0	1	3	6
	4	7	13	22

The output differences are , but the input steps are . Compare rates:

The constant rate per unit input shows a linear relationship: . Requiring equal raw output differences for unequal input steps would give the wrong answer.

Recognize functions in different representations

IN YOUR MVP PACKETMain task · pp. 13–18 · Problems 1–4, 7–8, 10–11, 16

Graphs, equations, tables, written descriptions, and recursive rules all express the same two patterns.

From a graph

A straight, nonvertical line represents a linear function. Compare the line in Problem 1 with the curved graph in Problem 11 of your packet. The line has the same slope from one location to another. The exponential-shaped curve becomes steeper as x increases: equal x-steps do not add the same amount, but its growth is consistent with a constant multiplication factor.

Don't classify a graph as exponential just because it curves. The curved graph in Problem 5 is neither linear nor exponential. Its shape bends downward rather than growing steeper like the increasing exponential graph in Problem 11.

From an equation

Worked example 4 — Recognize the equation

Compare , , and .

Equation	Classification and reason
	Linear: x is multiplied by 4; each extra unit of x adds 4 to y.
	Exponential: x is the exponent; each extra unit of x multiplies y by 4.
	Neither: x is squared, so equal x-steps do not give a constant added amount or a fixed multiplier.

In Problem 7, rearrange an equation such as before classifying: . This is linear.

From a written situation

Words	What to look for
“$25 more each week” or “$25 less each week”	Linear: a constant number added/subtracted
“Increases by 75% each year”	Exponential: multiply by each year
“Decreases by 1.5% each year”	Exponential: multiply by each year
“An object's height depends on its age”	Not enough for a fixed pattern; many actual growth processes are neither

From a recursive rule

A recursive rule tells you how to find a term from earlier terms. A starting value is essential.

Worked example 5 — One recursive rule is exponential; the other is neither

Rule A: , and . The first terms are . Each is twice the preceding term, so this is a geometric sequence (an exponential pattern on whole-number inputs).

Rule B: , , and . The first terms are . The differences are not constant, and neither are successive ratios. This is neither linear nor exponential.

Recognize “neither” and justify the choice

IN YOUR MVP PACKETMain task · pp. 14–18 · Problems 5–6, 9, 12–15

Use these examples to rule out both linear and exponential patterns without guessing from appearances.

For a relationship to be neither, you should be able to say what goes wrong with both patterns. A useful response contains two independent observations, not two ways of repeating the same claim.

Worked example 6 — A curved function that is neither

Suppose we have values .

The differences are , which are not constant; the ratios are , also not constant. Therefore it is neither linear nor exponential. This is a quadratic pattern; you do not need to learn how to solve quadratics for today's assignment.

Worked example 7 — Trip time versus speed

For a fixed trip of 120 miles at constant speed, time in hours is , where .

Speed (mph)	30	40	60	80
Time (hours)	4	3	2	1.5

The changes in time for equal 20-mph steps from 40 to 60 to 80 mph are hour and hour—different amounts. The ratios and are different too. The dependence on speed is neither linear nor exponential.

Packet clarification — Main task, Problem 14

“Each point on the graph is exactly 1/3 of the previous point” is not precise: a point has two coordinates and cannot simply be “one-third” of the preceding point in the sense intended here. The likely intended statement is: when the input increases by one unit, each new output is one-third of the previous output. With that assumption the outputs follow an exponential (geometric) pattern. Without equal input steps, this classification is not determined.

Packet clarification — Main task, Problem 9

A finite table of height and shoe-size measurements can show that the listed observations do not lie on a single linear or exponential pattern, but it cannot prove the form of every possible function outside those measurements. Classify the represented data as neither of the two familiar models; do not claim the table determines a unique rule for every height.

Checklist for two reasons

1. Describe the behavior (equal differences, equal factors, or neither) using actual values, the equation, or the graph. 2. Supply a different kind of evidence: a slope, a second calculation, a matching equation form, or an explanation of why the alternative pattern fails.

2. READY — Compare rates of change

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. 19 · Problems 1–5

Compare slopes in tables and line graphs, and interpret changes in real-life and exponential situations.

Rates in linear tables and graphs

A linear relationship has a single slope, so we can compare two lines by finding their change in y per one unit of x. Use the axes and scale, not simply the angle the line seems to make on the page.

READY worked example — Comparing two linear rates

Relationship A is recorded at and . Relationship B is given by .

Both are positive changes per one unit of x, and relationship B increases faster because . If both were decreasing, a more negative slope would mean a faster decrease in magnitude, even though it would be a smaller signed slope.

Rates in exponential situations

An exponential function does not have one constant additive rate. Its numerical change per unit input depends on where you measure it, even though the multiplier over equal input steps stays constant.

READY worked example — Comparing changes over the same interval

Compare and from to .

Over this interval, B increases by more. But to compare their rates later, measure changes over another specified interval; “which exponential grows faster?” cannot always be answered by comparing a single pair of slopes.

Packet clarification — Ready, Problems 2–3

Problem 2: Each curve gets steeper, so neither has a single constant slope. Use comparable intervals on the two graphs, and describe the visible comparison rather than claiming one permanent rate of change. Problem 3: Lee loses $25 each week; Jose pays half his remaining $50 debt each week. Jose pays $25 the first week, $12.50 the second, and $6.25 the third. The first-week changes are equal; after that, Lee's $25 weekly change exceeds Jose's weekly payment. The answer depends on the interval.

Shapes and repeated multiplication

For Ready Problem 4, the table's x-values rise by 4 as y rises by 2, giving a slope of y-unit per x-unit. The rhombus diagram builds more pieces as the figure number increases; identify the number added in each new figure, taking one figure as one input step.

For Ready Problem 5, compare the multiplier in with the multiplier in the Magic Pot story. If an object passes through a pot and doubles each time, growth depends on how many doubling stages are meant. Five consecutive doubling stages multiply a quantity by ; a single stage multiplies by 2.

Packet clarification — Ready, Problem 5

The Magic Pot description does not specify whether the five pots work in sequence, in parallel, or whether a “round” means one use of a single pot or all five. State the process you assume before comparing growth factors. Different procedures can create different multipliers.

Check your work — Ready 1–5 answer key

(b): its slope is 6, using (0, −3) and (1, 3); the table in (a) has slope 5.
(b) rises faster on the displayed positive-input portion. The grids have no numerical scales; this comparison assumes matching scales. Exponential rates vary with input.
Comparing weekly payment amounts: they tie at $25 in week 1; thereafter Lee pays more. Lee continues paying $25, while Jose’s successive payments halve. If comparing balances, specify whether “greater” means signed change or amount paid.
(b): rhombi increase by 2 per figure, versus $\frac12$ per x-unit in (a).
(a), under the intended comparison of factor 5 with factor 2. Having five pots does not specify how they are used; five successive doublings would instead multiply by 32.

3. SET — Classify equations, graphs, and stories

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 · p. 20 · Problems 6–11

Recognize a growth rate stated as a percentage, rewrite an equation, and distinguish a square-area pattern from an exponential one.

SET worked example — A percent decrease

A town starts with 8,000 people and decreases by 1.5% per year. Each year 98.5% of the previous year's population remains:

This is exponential: the repeated factor is , not a constant subtraction of 1.5 people or a constant number of residents.

SET worked example — Commission and a base salary

Suppose a worker earns a $30,000 base salary plus 4.25% of total sales . Total pay is

This is linear because each additional dollar of sales adds exactly of commission; the coefficient of s is constant. The percentage is of sales, not of the worker's previous earnings. Seeing a percent sign does not automatically mean exponential.

For Set Problem 8, rearrange to get : linear.

For Set Problem 9, the daily gifts in “The 12 Days of Christmas” add a new set of gifts each day. The total received on each day follows ; the differences and ratios are not constant. This is neither linear nor exponential. Be careful not to confuse daily gifts with the cumulative total through that day.

For Set Problem 11, the area values for side lengths 1, 2, 3, 4 are 1, 4, 9, 16. Equal increases in side length do not give equal area increases or equal multiplication factors. This is neither, not exponential just because the outputs increase.

Check your work — Set 6–11 answer key

Exponential: annual multiplier $0.985$.
Linear: earnings $E=30000+0.0425s$, with sales s as input.
Linear: $y=-\frac34x-\frac34$.
Neither: daily gift totals are $1,3,6,10,\ldots$, with increasing differences.
Neither an unshifted exponential nor linear: the curved graph approaches a nonzero height to the left and crosses the x-axis.
Neither: area $A=s^2$ is quadratic.

4. GO — Geometric means and slope

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 · pp. 20–21 · Problems 12–22

Fill missing terms in geometric sequences (12–16), then calculate slopes from tables, coordinate pairs, and a graph (17–22).

Geometric means: Find the missing ratio first

A geometric sequence has the same multiplier between every consecutive pair of terms. If the first term is , then

To fill missing terms, count the number of jumps from the known first term to the known last term. If four jumps take to , then . Find possible r-values and multiply successively.

GO Problem A — Straightforward geometric means

Find the three missing terms in a geometric sequence beginning at 4 and ending at 324 after four steps, assuming a positive ratio.

Multiply by 3 four times: . Verify .

MVP clarification — Section 2.3, Go, Problems 12–16

Module 1, Section 1.10 taught that some geometric sequences can have more than one valid solution. A geometric common ratio can be negative, so it is worth checking both signs when you find missing terms.

Problem 12: The first term is 2 and the fifth term is 162. Both sequences satisfy the given information:

2, 6, 18, 54, 162 — common ratio 3.
2, −6, 18, −54, 162 — common ratio −3.

Problem 13 is different: its first term is 1/9 and its fourth term is −3. Because these terms are three multiplication steps apart, the real common ratio is uniquely −3.

Remember: when finding missing geometric terms, check whether positive and negative common ratios both work. This is a reminder of mathematics introduced earlier in MVP, not a new restriction.

GO Problem B — More challenging: A negative ratio

A geometric sequence has and . Find .

Successively multiply by : . Because the endpoints are three steps apart and the ratio must have a negative cube, here the real ratio is uniquely .

GO Problem C — Unknown letters in geometric sequences

Suppose and , with . Then

For real values, and both work. If a positive ratio is stipulated, use . The two possible sequences are and . If , the given endpoints do not determine a unique ratio at all.

Rates of change from tables and points

The slope between two points is the change in the vertical variable divided by the change in the horizontal variable:

When x-steps are unequal, divide the y-difference by the actual x-difference. For a straight line this quotient is the same for any two distinct points.

GO Problem D — Straightforward slope from a table

	-4	-1	2
	9	3	-3

Check the next interval: . Same result.

GO Problem E — More challenging slope from separated points

A line contains and . Find its slope.

Note how subtracting a negative in the denominator becomes addition: .

For Go Problem 21: read two points on the drawn line whose coordinates are clear (prefer grid intersections), then use rise over run. You do not need to redraw the packet's existing graph.

Check your work — Go 12–22 answer key

Missing terms: $6,18,54$ (ratio 3). A negative ratio also gives $-6,18,-54$.
Missing terms: $-\frac13,1,9$ (ratio $-3$).
Missing terms: $5,2.5,1.25$ (ratio $\frac12$); ratio $-\frac12$ instead gives $-5,2.5,-1.25$.
$gz,gz^2,gz^3$, using ratio z; ratio $-z$ gives $-gz,gz^2,-gz^3$ as another possibility.
$-9,-27,-81$ (ratio 3); ratio $-3$ instead gives $9,-27,81$.
No single constant slope as printed: consecutive slopes are $-\frac72,-\frac72,-\frac{13}{4}$. The apparent intended slope is $-\frac72$, which requires the final table output to be −6.5 rather than −6.
$2$.
$2$.
$4$.
$\frac14$.
$2$.

Additional Learning Resources

These resources explain the same skills from a more traditional angle. Use the suggested parts; some sources continue into material beyond today’s lesson.

Resource and direct link	What it covers / why use it	Scope / What to skip
Khan Academy — Exponential vs. linear growth (video)	Direct match for classifying equal differences versus equal ratios in tables.	Close match. Watch the full short comparison.
Paul's Online Math Notes — Exponential Functions and practice with solutions	Clear written explanation of , graph behavior, and worked values. Practice problems include direct function evaluations.	Some extra material. Use the definition, first graphing example, and basic function-value practice. Skip sections introducing , transformed exponential graphs, and more advanced equations.
Khan Academy — Geometric sequences review	Extra help with Go 12–16: common ratio, explicit and recursive formulas, and missing terms.	Close match. Focus on common ratios, including negative ratios.
Professor Leonard — Intermediate Algebra Lecture 12.3: Graphing and Solving Exponential Functions	Optional longer classroom-style review of exponential equations and graphs.	Significantly broader. Use the introductory discussion of exponential form and graphs; skip later equation-solving techniques not assigned in Section 2.3. Specific timestamps could not be verified.

Teacher's Review

The main task appropriately practices classification across representations. The following issues deserve specific clarification rather than a general criticism of the instructional approach.

Main task, Problem 14 — “each point” is not a precise growth rule

The task describes each point as one-third of the previous point, but does not identify whether the y-values are being multiplied, how the inputs progress, or what “previous” means in the graph. For a standard geometric or exponential interpretation, it must state that outputs are multiplied by one-third over equal input steps. Without that condition, classification as exponential is not established.

READY Problems 2–3 — Rate-of-change comparison is underspecified

Exponential functions have changing absolute rates, so “greater rate of change” requires a specified input interval or an indicated comparison point. Problem 3 also compares a constant weekly $25 deduction with a debt repayment that halves each week: the first changes are tied at $25 and later repayments are smaller. The question should specify the interval and whether “greater” refers to signed change or the magnitude of the decrease.

READY Problem 5 — Ambiguous Magic Pot procedure

Five pots are mentioned without specifying whether the doubling is sequential, parallel, or measured per one complete round. Consequently, the effective factor cannot be uniquely compared with without identifying the process.

Curriculum Coverage

MVP assignment	Skills assessed	Where covered here
Main, 1–5	Graph or table shape; linear, exponential, neither; equal differences or ratios	In class, worked examples 1–2 and 4–6
Main, 6–12	Real contexts; linear equations; ordered data; exponential curves; inverse speed/time	In class, worked example 7
Main, 13–16	Square-power expression; verbal ratio; recursive rules	In class, worked examples 4–5 and Problem 14 clarification
Ready, 1–5	Slope comparison, changing exponential rates, growth contexts, patterns in figures	Section 2, Ready examples and clarifications
Set, 6–11	Percent changes, commission, rearranging equations, other growth patterns	Section 3, Set worked examples
Go, 12–16	Geometric means, common ratios, including negative ratios and symbolic terms	Section 4, Problems A–C and red clarification
Go, 17–22	Slope from tables, point pairs, a graph	Section 4, Problems D–E

The examples and practice explanations above support the mathematical skills required for the listed packet problems; they do not reproduce the worksheet's complete solutions.

Last updated October 13, 2026.