Last updated October 13, 2026.

Secondary Math I · Module 2 · Linear & Exponential Functions

2.6 — Form Follows Function

Supplement for Mathematics Vision Project (MVP) curriculum · Secondary Math I, Module 2, Section 2.6
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 what information a situation gives you; select an efficient linear or exponential equation form; use that form to graph a function; compare linear and exponential models; and justify the steps for solving one-step equations.

1. In class: A Practice Understanding Task

Choose the equation form that fits the information

IN YOUR MVP PACKETMain task · printed p. 35 · reference tables

Review the linear and exponential forms already introduced in this module. The challenge today is choosing, rather than deriving, the right tool.

You have learned several ways to describe the same type of change. There is no single form that is always best. Choose a form that lets you use the information given without first finding unnecessary quantities.

Linear change: add or subtract the same amount

Suppose a quantity begins at 6 when the input is zero and increases by 2 whenever the input increases by one. The slope is 2. The rule is

This is called slope-intercept form, conventionally written , because it shows the slope and the y-intercept .

If instead you know that a line passes through with slope 2, you can start from that point:

This is point-slope form, written . Distributing shows that both equations describe the same line: .

If the input counts whole-number steps, you can also use a recursive rule: start with , then for . A recursive rule needs its starting value and the step-to-step rule.

Exponential change: multiply by the same factor

Suppose a quantity begins at 6 and triples each step. Its explicit rule is

The coefficient 6 gives the value at input 0. The base 3 gives the multiplier per one input step. For a discrete sequence you can instead write and .

A useful clarification — Ready, Problem 8

To find the y-intercept from an equation, set x = 0. You may have learned to read the y-intercept directly from slope-intercept form, y = mx + b. That works because setting x = 0 makes the mx term zero, leaving y = b. Reading b is just a shortcut for the substitution.

For an exponential equation, y = a · r^x, do exactly the same thing: set x = 0. Since r⁰ = 1 for a positive base, the result is y = a. The procedure is identical; the equations simplify differently.

Choose efficiently

Linear: starting value at x=0 and slope → slope-intercept; another known point and slope → point-slope; initial term and stepwise difference → recursive if you need only the next few terms. Exponential: starting value and multiplier → explicit; first known term and common ratio → either recursive or shifted explicit, depending on the question. For a far-away term, explicit form avoids applying the recursion many times.

These are the forms in the packet's toolbox, not five unrelated kinds of mathematics. A linear sequence is arithmetic when restricted to consecutive integer inputs; an exponential sequence is geometric when restricted to consecutive integer inputs.

Worked example 1 — Decide between the two linear forms

A line has slope and passes through . No y-intercept is given, so use point-slope form:

If you want slope-intercept form, simplify: . Both describe the same line. Starting from avoids the extra calculation of the intercept.

Turn descriptions, graphs, and tables into equations

IN YOUR MVP PACKETMain task · printed pp. 36–38 · Problems 1–5

Choose a function type, write an equation, name its form, and classify a reasonable domain. The packet prints two separate problems numbered 3.

Read the units before choosing the rule

First decide what the input counts and what the output measures. Then ask whether the change is additive or multiplicative. Finally, decide which form makes the given information easiest to use.

Worked example 2 — A fixed payment plus an amount per item

Joe earns per month plus for each vacuum sold. Let n be the number sold and I(n) the monthly income in dollars.

Here 500 is the starting income at n=0 and 20 is the increase per vacuum. This is linear, slope-intercept form. Because n is a count of vacuums, the contextual domain is : discrete, even though the formula is linear.

Worked example 3 — A geometric sequence whose first index is 1

Suppose the first term of a geometric sequence is −3 and the common ratio is 5. Because the first term occurs at n=1, write:

The first terms are . Starting from , the recursive rule is . This is a discrete geometric sequence; its negative outputs do not make it “neither.”

Watch the index: The expression starts at −3 when n=0, not when n=1. Either convention works if stated consistently.

Worked example 4 — Identify the equation of the graphed line

The printed graph in the second Problem 3 passes through and . The slope is

Using the y-intercept immediately gives . A straight, unbroken line represents a continuous linear function on the depicted real-input interval.

Worked example 5 — Convert percent growth to an exponential model

The population of Java Hot Springs begins at 35,000 in 2003 and increases by about 2.4% annually. “Increase by 2.4%” means each year the model retains 100% and adds 2.4%, so its multiplier is .

This is exponential growth. At t=1 the model predicts . The formula can be evaluated at whole-number years; a continuous-time extension is also possible when modeling an evolving population approximately.

Worked example 6 — A table with a constant additive change

The plant-height table reports cm in weeks 1 through 5. Each consecutive increase is cm.

Begin with the known point (week 1, 1.7 cm):

This is a linear model. The measurements exist only at weekly checkpoints, but actual plant height varies between measurements, so a continuous-time model is reasonable. The formula is an idealized model; measured biological growth need not be perfectly linear between observations.

Packet clarificationContinuous versus discrete: choose the input domain

For the sequence in main Problem 3, the input is the term number, so discrete is clear. For the town-population and plant-growth contexts, the packet does not explicitly require either annual/weekly-only inputs or every real elapsed time. Observed data at integer times do not by themselves force the underlying model to be discrete. State your modeling choice: integer time points for a sequence of observations, or elapsed real time for a continuous model. People are counted in whole numbers, but that does not make time itself a discrete input.

2. Ready — Compare linear and exponential 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 packet: Section 2.6 · Ready · pp. 40–41 · Problems 1–8.

Use the equation-form reminders above to compare constant addition with constant multiplication.

Check your work — Ready 1–8 answer key

$y=4+3x$: linear growth. $y=4\cdot3^x$: exponential growth.
Linear model: arithmetic sequence. Exponential model: geometric sequence.
Sample points: linear $(0,4),(1,7),(2,10),(3,13)$; exponential $(0,4),(1,12),(2,36),(3,108)$.
Linear: constant rate 3. Exponential: multiplier 3 for each unit increase in x; its additive rate is not constant.
Both increase and pass through $(0,4)$. The linear graph is straight; the exponential graph curves and becomes steeper. Plots are not shown.
Both: $(0,4)$.
(a) $(0,0)$. (b) $(0,1)$.
Set x to zero in either equation. For $y=mx+b$, this gives b; for $y=ab^x$, $b^0=1$, so it gives a.

3. SET — Select the efficient form for each situation

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 · printed pp. 41–42 · Problems 9–14

Choose equation forms in financial, folding, weight-change, and walking contexts, then graph an exponential decay rule and a point-slope linear rule.

Use a known point and slope

Reminder: point-slope form

When you know the slope m and one point on a line, you can write its equation directly:

This uses a point you already know. You do not have to find the y-intercept first.

Worked example 7 — Monthly savings from a known point and slope

Jasmine deposits $175 each month. After 10 months, her savings balance is $2,475. Write an equation for her balance after t months.

Step 1: Identify the slope and a point. The balance grows by $175 per month, so the slope is . After 10 months the balance is $2,475, so we know the point .

Step 2: Use point-slope form. Substitute the known point and slope:

Add 2,475 to both sides to write an equation for the balance:

The expression t − 10 measures how many months have passed since month 10. The equation starts with the known $2,475 balance, then adds $175 for each additional month. It also works for earlier months.

Step 3: Check the known balance. At month 10:

If you expand and simplify, the same equation becomes . This shows that Jasmine's starting balance was $725, but you do not need to calculate it first to write the equation.

Worked example 8 — Folding paper: near-term recursion or far-term explicit

The number of rectangles doubles with each fold: at folds 1, 2, 3, and 4 the table gives 2, 4, 8, and 16 rectangles.

A recurrence is , then . This is convenient to find the next value: .

For a distant term, use an explicit rule: . At fold 14, .

Worked example 9 — Constant decrease in a measurement

Starting from 90 kg, a model that loses 1.2 kg each week is linear:

This uses the starting value and rate directly in slope-intercept form. Here t=0 represents now. The rule is intended for the several weeks described, not for arbitrarily large t or negative weights.

Worked example 10 — Use a known point and slope

Janet increases the distance she walks by 10 meters each day. On day 20, she walks 800 meters. Write an equation for her daily walking distance and find the distances on days 21 and 60.

Step 1: Identify the slope and a point. The slope is because she walks 10 meters farther each day. The known point is because on day 20 she walks 800 meters.

Step 2: Use point-slope form. Let n be the day number and D(n) the distance in meters. Substitute the point and the slope:

Add 800 to both sides to get an equation for the distance:

The (n − 20) counts how many days have passed since day 20. Start with 800 meters on that day, then add 10 meters for each day after it.

Step 3: Check the given point, then answer the questions.

Answers: 810 meters on day 21 and 1,200 meters on day 60.

You can also simplify this to , the slope-intercept form of this same line. Both equations describe the same pattern; point-slope form avoids calculating the day-zero value first. Use whole-number day inputs for this sequence.

Graph the two Set equations from their visible information

Worked example 11 — Exponential decay, Set Problem 13

The given rule is . Its initial value at n=0 is 8, and it halves each time n increases by one.

n	0	1	2	3	4
y	8	4	2	1	0.5

Plot four or five points; if n is a real-valued input, draw a smooth positive decay curve approaching the x-axis without touching it. If n represents sequence indices, leave the points discrete.

Worked example 12 — Point-slope, Set Problem 14

The given rule reveals point and slope 5. Four points are . Graph the straight line through them, using a vertical scale that includes −6 through 9.

Check your work — Set 9–14 answer key

Point-slope: $B(m)=175(m-10)+2475$, equivalently $B(m)=175m+725$.
(A) Recursive: $R(1)=2,\ R(n+1)=2R(n)$; 5 folds give 32 rectangles. (B) Explicit: $R(n)=2^n$; 14 folds give 16,384 rectangles.
Slope-intercept: $W(t)=90-1.2t$, with elapsed weeks t.
Day 21: 810 m (recursive step). Day 60: 1200 m, using point-slope $D(n)=10(n-20)+800$.
Exponential decay, initial value 8 and multiplier $\frac12$. Sample points: $(0,8),(1,4),(2,2),(3,1)$. Plot not shown.
Slope 5; known point $(2,-6)$. Sample points: $(0,-16),(2,-6),(3,-1),(4,4)$. Plot not shown.

4. GO — Solve one-step equations and justify each step

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 · printed p. 43 · Problems 15–20

Use the additive and multiplicative properties of equality to solve equations. Justify the operation and simplification, not just the final answer.

A solution is a number that makes an equation true. To keep an equation balanced, apply the same operation to both sides.

Properties of equality

Additive property: If A = B, then A + c = B + c. Subtracting c is adding −c. Multiplicative property: If A = B, then cA = cB for any c; to undo multiplication by a nonzero number, multiply both sides by its reciprocal. Identities: A + 0 = A and 1 · A = A.

The goal is to isolate the variable by undoing the operation attached to it. State the property applied, then perform the arithmetic, and finally substitute the answer to check.

Worked example 13 — An addition equation, with justification

Solve :

Equation	Reason
	Given
	Additive property of equality: add 10 to both sides
	Simplify −10 + 10 = 0 and 2 + 10 = 12
	Additive identity: x + 0 = x

Check: , so the solution is correct.

Worked example 14 — A negative multiplier, with justification

Solve . Multiply both sides by the reciprocal of −3, namely :

Equation	Reason
	Given
	Multiplicative property of equality
	Multiply the coefficients;
	Multiplicative identity: 1x = x

Check: .

Worked example 15 — The variable is on the right

Solve . Add −11 to both sides:

By the additive property of equality and the additive identity, the solution is . Checking gives . You do not need to move the variable to the left first.

Other packet cases: For , multiply both sides by , giving x=5; for , subtract 6, giving x=4; for , multiply both sides by , giving x=3. Each uses the same equality principle.

Check your work — Go 15–20 answer key

$x=5$; multiply both sides by $\frac13$ (multiplicative property).
$x=12$; add 10 to both sides (additive property).
$x=-27$; subtract 11 from both sides (additive property).
$x=4$; subtract 6 from both sides (additive property).
$x=3$; multiply both sides by $\frac16$ (multiplicative property).
$x=-\frac23$; multiply both sides by $-\frac13$ (multiplicative property).

Additional Learning Resources

Specific resource and link	Why it helps	Scope / What to skip
Paul's Online Math Notes — Lines	Detailed explanation of slope-intercept and point-slope forms, and graphing from their structure. Supports main task and Set 9, 12, 14.	Some extra material. Focus on slope and the two equation forms; skip parallel/perpendicular lines and vertical-line details unless reviewing.
Paul's Online Math Notes — Lines practice with solutions	Short practice problems on finding slope and both forms of a line; includes links to full solutions.	Some extra material. Use practice problems 1–7, especially 3–5; skip problems 8–9 on parallel/perpendicular lines.
Professor Leonard — How to Use Slope-Intercept Form (TTP Video 41)	Optional lecture-style review of how an equation's slope and intercept help graph a line efficiently.	Close match for graphing linear equations. Focus on slope-intercept graphs; timestamps were not verified. It does not address exponential models or one-step equation justifications.
Khan Academy — Linear vs. exponential growth: from data	Video models data by checking whether outputs change additively or multiplicatively; supports the main task and Ready comparison.	Close match. Focus on comparing and modeling data; skip any extension beyond simple linear/exponential rules if unfamiliar.
Khan Academy — Graphing exponential growth and decay (practice)	Graphing practice for exponential equations, supporting main task 7 and Set 13.	Close match. Applies to continuous curves; remember the MVP task can also use discrete integer-index sequences.
Khan Academy — One-step equations review	Written walkthrough of inverse operations and balancing both sides of an equation, supporting Go 15–20.	Close match. Supplement your written solution by stating the equality property at each step, as the packet requires.

Teacher's Review of the Original MVP Packet

Overall: Section 2.6 is a useful consolidation lesson. Unlike many discovery-based tasks, it names the forms directly and asks students to explain why a chosen form is efficient. The teacher guide explicitly supports comparing different valid methods.

Where a short clarification would help

Domain classifications: For main task Problems 4–5 and the exponential graph in Problem 7, the packet sometimes asks for or implies a continuous/discrete classification without stating the permitted input domain. Population is counted in whole people but time may be continuous; a plant's height changes between weekly observations; an equation such as can be given a real or integer domain. Specify the intended model or accept a justified domain choice. This is a modeling clarification, not an indication that the equations are wrong.

Minor navigation issue: Two consecutive main-task problems are both numbered 3. Refer to them as the “geometric sequence Problem 3” and the “graphed-line Problem 3.”

The remainder of the packet is mathematically coherent. Set targets efficient representation; Go explicitly teaches equality properties and asks students to justify their use.

Topics Covered in This Supplement

This is a selective guide to topics that benefit from additional explanation, not a replacement for every exercise in the MVP packet.

MVP assignment	Skills taught	Where in this lesson
Main task, p. 35	Recognize slope-intercept, point-slope, linear recursive, exponential explicit, exponential recursive forms	In class, example 1
Main Problems 1–2, p. 36	Model a linear pay situation and a line through a point; discrete versus continuous inputs	In class, examples 1–2
Main Problems 3 (sequence and graph) and 4, p. 37	First-term indexing, graphed-line slope/intercept, percent growth	In class, examples 3–5
Main Problem 5, p. 38	Find linear model from time-series data, distinguish sampling from domain	In class, example 6
Set Problems 9–12, pp. 41–42	Choose efficient formulas for savings, folding, changes in measured weight, daily walking	Section 2, examples 7–10
Set Problems 13–14, p. 42	Graph exponential decay and point-slope linear equations, give four ordered pairs	Section 2, examples 11–12
Go Problems 15–20, p. 43	Additive and multiplicative equality properties, inverse operations, identities, solution checks	Section 3, examples 13–15 and additional cases

Page numbers refer to pages printed in the MVP packet, rather than PDF-viewer page numbers. This independent supplement is an instructional guide: where it illustrates specific packet problems, it explains why the equations and steps work.

Last updated October 13, 2026.