How do I decide linear versus exponential?

Look at the change pattern: constant additive change indicates a linear model; constant multiplicative or percent change indicates an exponential model. Check tables and discrete ratios across equal intervals to guide selection.

What is the best way to teach graph interpretation?

Use paired examples with identical starting values but different change rules, emphasize axis labeling and scale, and have students predict future values before plotting to reinforce model differences.

Can I collect student submissions electronically?

Yes. Electronic collection is acceptable; ensure the platform preserves timestamps and audit logs. For signed parental permissions, follow ESIGN Act and UETA requirements for intent and consent.

Are there privacy rules for student data?

Yes. Student education records are protected by FERPA; any health-related student data may trigger HIPAA. Limit personally identifiable information in public examples and secure access to submissions.

How should teachers grade graph uploads?

Use a rubric that scores axis labels, scale accuracy, plotted points, and interpretation. Require uploaded images to be legible and provide guidance on acceptable file formats.

What if a parent requests a paper copy?

Provide a printable version consistent with the electronic lesson. For signed permissions, offer both written and ESIGN-compliant electronic consent options per district policy.

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Lesson 3: Linear Versus Exponential Functions

Lesson 3: Linear Versus Exponential Functions

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Lesson 3: Linear Versus Exponential Functions

Name: Period: Date:

Learning Targets:

• I can distinguish between situations that can be modeled with linear functions and with exponential functions.

• I can state that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors/ratios over equal intervals.

Linear and exponential functions share many characteristics. This is because they are based on two different, but similar, sets of principles.

LINEAR VERSUS EXPONENTIAL

Linear functions are based on the same amount

The slope (m) – Constant rate of change – Common difference

Exponential functions are based on by the same amount

The base (b) – Growth or decay factor – Common Ratio

Example #1: The two tables below represent a linear function and an exponential function.

Part 1: Which type is each function below? Explain how you arrive at your answer.

TABLE 1

x	0	1	2	3	4
y	5	10	20	40	80

Type

TABLE 2

x	0	1	2	3	4
y	8	11	14	17	20

Type

Part 2: Find equations in standard form for each of the functions from Example #1.

Table 1 Equation:

Table 2 Equation:

Part 3: Sketch the graph of each equation

Example 2: Consider the linear function y = 20x + 5 and the exponential function y = 5(2)^x. Make a sketch of their graphs. Which one of these grows faster?

Example 3: Which of the following functions would best describe the data in the table?

(1) y = 10x + 2

(2) y = 8x + 2

(3) y = 5(2)^x

(4) y = 2(5)^x

x	0	1	2	3	4
y	2	10	50	250	1250

Example 4: Find the equation of the exponential function, in y = a(b)^x form for the function given in the table below.

x	0	1	2	3	4
y	10	30	90	270	810

Linear functions grow while exponential functions grow

Example 5: Write an equation of the function represented in the table below.

x	-1	0	1	2	3	4
f(x)	2/3	2	6	18	54	162

Type Equation

Example 6: Write an equation of the function represented in the table below.

x	-3	-2	-1	0	1	2
f(x)	5.5	5	4.5	4	3.5	3

Type Equation

REASONING. You can determine the equation of a line or the equation of an exponential given any two points that lie on these curves. In this exercise we will pick two special points. Consider the points (0, 5) and (1, 15)

a. Write the equation of the line that passes between these two points in y = mx + b form.

b. Write the equation of the exponential that passes between these two points in

c. Using your calculator, sketch the two graphs on the axes below. Label with their equations

d. Is it fair to say that an exponential function always grows faster than a linear?

e. What can we say about an increasing exponential function when compared with an increasing linear function?

f. What is the difference between the way a linear function increases and the way an exponential function increases?

Lesson Summary

• Linear function of the form L(x) = mx + k grows additively by the same number over equal length intervals.

• Exponential function of the form f(x) = ab^x grows multiplicatively by the same factor b over equal length intervals.

• An increasing exponential function will eventually exceed any linear function. Sometimes this is not apparent in a graph displayed on a graphing calculator because the graphing window does not show enough of the graphs for us to see the sharp rise of the exponential function in contrast with the linear function.

Problem Set

1. Look at the tables below. Which tables show a linear function and which show an exponential function?

Table 1

x	y
0	2
1	8
2	32
3	128
4	512

Type

Equation:

Table 2

x	y
0	20
1	17
2	14
3	11
4	8

Type

Equation:

Table 3

x	y
0	81
1	27
2	9
3	3
4	1

Type

Equation:

2. Determine the function type: linear or exponential that can be used to model the data set and then write an equation.

Table 1

x	y
0	1
1	5
2	25
3	125
4	625

Type

Equation:

Table 2

x	y
0	300
1	150
2	75
3	37.5
4	18.75

Type

Equation:

Table 3

x	y
0	-6
1	2
2	10
3	18
4	26

Type

Equation:

3. Consider the points A(0,9) and B(1,3).

a. Write the equation of the line that passes between these two points in y = mx + b form.

b. Write the equation of the exponential that passes between these two points in

c. Using your calculator, sketch the two graphs on the axes below. Label with their equations

4. Write an equation of the function represented in the table below.

x	-3	-2	-1	0	1	2
f(x)	48	24	12	6	3	3/2

Type

Equation

5. A lab researcher records the growth of the population of a yeast colony and finds that the population doubles every hour.

a. Complete the researcher’s table of data:

Hours into study	0	1	2	3	4
Yeast colony population (thousands)	5

b. What is the exponential function that models the growth of the colony’s population?

6. Write an equation of each function represented in the table below.

x	g(x)
-3	-7
-2	-3
-1	1
0	5
1	9
2	13

g(x) =

x	h(x)
-1	1.25
0	5
1	20
2	80
3	320
4	1280

h(x) =

How does the y-intercept of the function g(x) compare to the y-intercept of h(x)?

How does the rate of change of the function g(x) compare to the rate of change of h(x)?

7. Write an exponential equation for the graph shown below. Regents 01322015

Equation:

Explain how you determined the equation.

What Lesson 3 covers and why it matters

Lesson 3: Linear Versus Exponential Functions is a classroom unit that explains how linear relationships differ from exponential growth and decay, and how to model each with equations and graphs. The lesson covers slope and intercept for linear models, base and rate for exponential models, graph interpretation, and real-world examples such as interest, population change, and decay. Instruction includes worked examples, guided practice, graphing activities, and formative assessment items designed to build conceptual understanding and procedural fluency for secondary mathematics learners.

Why understanding these functions improves modeling skills

Mastering linear and exponential models helps students select appropriate mathematical tools for prediction, interpretation, and communication in STEM and finance contexts. It builds transferable reasoning for real-world data and prepares students for advanced algebra, statistics, and science courses.

Who benefits from Lesson 3

This lesson supports multiple classroom roles and learning levels.

Secondary math teachers who need a structured unit with examples, assessments, and classroom activities for grades 8–11.
Students learning to distinguish proportional linear change from multiplicative exponential change using graphs and equations.
Curriculum designers and instructional coaches creating aligned sequences that integrate practice, assessment, and remediation.

Materials can be adapted for accelerated classes or scaffolded for learners needing additional support.

Core components included in Lesson 3

The lesson is designed as a modular unit that combines explanation, worked examples, practice items, assessment probes, visual graphing tasks, and extension activities.

Learning Objectives

Clear measurable targets such as identifying slope, interpreting rate of change, and distinguishing exponential growth or decay contexts.

Key Vocabulary

Terms like slope, intercept, rate, base, exponential factor, and doubling time with student-friendly definitions and examples.

Worked Examples

Step-by-step problems showing algebraic derivation, table creation, and graph construction for both linear and exponential cases.

Practice Problems

A mix of procedural and conceptual questions, including answer keys and common error notes for rapid feedback.

Assessment Items

Formative quizzes and a summative problem set with rubrics and exemplar solutions to evaluate mastery.

Extensions

Challenge tasks connecting exponential models to compound interest, population dynamics, and basic differential reasoning.

Step-by-step: teaching and completing Lesson 3

Follow these steps for lesson delivery, student completion, and evaluation.

01

Preview Concepts: Introduce slope and exponential base with simple examples.
02

Model Comparison: Compare linear tables/graphs with exponential counterparts.
03

Practice Problems: Assign mixed problems, include graphing tasks.
04

Formative Check: Use short quiz to assess understanding and reteach as needed.

Digital delivery workflow for the lesson

This flow shows how content moves from assignment to grading when delivered electronically.

Assign: Teacher uploads lesson and sets due date in LMS or document platform.
Complete: Students answer problems, upload graphs, and submit electronically.
Review: Teacher grades using rubric, leaves feedback, and records scores.
Return: Students receive grades and targeted remediation suggestions.

Recommended online settings for classroom deployment

Configure these settings when you publish the lesson to an LMS or document-signing workflow.

Field	Configuration
Platform	LMS (Canvas, Google Classroom) or document platform with eSubmission.
Authentication	Student account login or single-sign-on (SSO) for identity assurance.
Submission Type	File upload (PNG/PDF) and typed response fields enabled.
Auto-Grading	Use auto-graded items for objective questions; manual for graph evaluation.

Technical requirements and integrations

Ensure the chosen platform supports file uploads, timestamps, and role-based access controls.

File Formats: PDF, PNG preferred.
Integrations: Works with Google Workspace and Microsoft 365 integrations.
Authentication: SSO or school-managed accounts.

For electronic signature or official eSubmission needs, platforms that integrate with document eSignature services and provide audit logs are recommended to maintain authentication and retention records.

Typical scheduling milestones for the lesson

Use these dates to plan instruction, practice, and assessment windows within a unit.

Lesson Release Date:

Day 1: Teacher presents concept and examples.

Practice Deadline:

Day 2–3: Students complete practice problems.

Assessment Window:

Day 4: Formative quiz or in-class assessment.

Feedback Return:

Within 3 business days after assessment grading.

Extension Activity:

Optional week-long challenge or project.

Key milestones from assignment to mastery

A sequential view of primary stages across the lesson lifecycle.

01

Lesson Assigned

Teacher publishes materials and deadline is set.

02

Student Practice

Students complete guided and independent exercises.

03

Assessment Administered

Quiz or problem set evaluated for mastery.

04

Review & Remediation

Targeted reteaching for identified gaps.

Common misunderstandings to anticipate

Interpreting slope as constant percent change rather than constant additive change, which leads to misclassifying exponential situations.
Incorrect graph scaling that obscures curvature of exponential growth or compresses linear trends, causing false interpretation.
Failing to convert rate language into proper parameters (e.g., 'doubles every three periods' versus misapplied linear increment).
Skipping units and labels on axes, which creates ambiguity when comparing growth rates across contexts.

Consequences of errors in assessment or record handling

Grade Impact: Lowered score or incomplete credit.

Invalid Assessment: Results may be unusable for mastery decisions.

Miscommunication: Students misapply models to real problems.

Data Privacy: FERPA protections apply to student records.

Compliance Risk: HIPAA concerns for health-related student data.

Retention Issues: Improper storage may violate recordkeeping rules.

Recordkeeping and security considerations

Encryption: TLS 1.2/1.3 in transit, AES-256 at rest

Access Controls: Role-based permissions and SSO recommended

Audit Logs: Maintain timestamps and activity history

FERPA: Protect student education records

HIPAA: BAA required for protected health data

Backup: Regular secure backups and versioning

How Lesson 3 differs from related materials

Compare Lesson 3 to a short worksheet and a unit assessment to choose the right resource for instruction.

Criteria	Lesson 3	Worksheet	Unit Assessment
Purpose	instruction	practice	evaluation
Interactivity	high	medium	low
Typical Length	multiple class periods	single page	full unit
Delivery Mode	digital or print	print or digital	digital

Typical eSignature vendor pricing and feature overview relevant to educational eSubmission

Price and feature choices affect how you collect signed permissions, submissions, or acknowledgements; signNow is listed first for comparison.

	signNow	DocuSign	Adobe Sign	PandaDoc	HelloSign
Starting Price	$8/user/mo	$15/user/mo	$14/user/mo	$19/user/mo	$15/user/mo
Free Trial	7-day free trial	Varies by plan	Varies by plan	Varies by plan	Varies by plan
Bulk Send	Yes	Yes	Yes	Yes	Depends on plan
Audit Trail	Yes	Yes	Yes	Yes	Yes
HIPAA Compliant	Yes	Yes	Yes	No	No
Envelope Cap	No cap	100 envelopes/user/year	Varies by plan	Varies by plan	Varies by plan

Two classroom examples illustrating application

Practical scenarios show how to use Lesson 3 for instruction and applied problems.

High School Algebra Class

Students model monthly savings as linear versus compound savings as exponential

Teacher compares tables and graphs for both
The class discusses when each model fits, then completes a short quiz to demonstrate correct model selection and interpretation.

University Epidemiology Module

Learners examine infection counts to contrast linear case increases with exponential outbreaks

Instructor adds mitigation factor to convert exponential to sub-exponential growth
Students perform parameter estimation and write a short explanation of public health implications.

Typical educators and designers who prepare this lesson

High School Teacher

A mathematics teacher planning a two-week unit uses the lesson to teach model selection, scaffold graphing skills, and assess student readiness for algebra II. They adapt practice sets, grade tasks, and provide targeted remediation based on formative checks.

Curriculum Designer

An instructional designer creates aligned resources and rubrics, ensures standards coverage, and packages digital assets for LMS deployment, including auto-graded items and rubric-based teacher scoring guides.

Frequently asked questions about Lesson 3 and digital delivery

Answers to common questions about concept clarity, delivery, assessment, and legal considerations for electronic records.

Need help? Contact support

How do I decide linear versus exponential?
Look at the change pattern: constant additive change indicates a linear model; constant multiplicative or percent change indicates an exponential model. Check tables and discrete ratios across equal intervals to guide selection.
What is the best way to teach graph interpretation?
Use paired examples with identical starting values but different change rules, emphasize axis labeling and scale, and have students predict future values before plotting to reinforce model differences.
Can I collect student submissions electronically?
Yes. Electronic collection is acceptable; ensure the platform preserves timestamps and audit logs. For signed parental permissions, follow ESIGN Act and UETA requirements for intent and consent.
Are there privacy rules for student data?
Yes. Student education records are protected by FERPA; any health-related student data may trigger HIPAA. Limit personally identifiable information in public examples and secure access to submissions.
How should teachers grade graph uploads?
Use a rubric that scores axis labels, scale accuracy, plotted points, and interpretation. Require uploaded images to be legible and provide guidance on acceptable file formats.
What if a parent requests a paper copy?
Provide a printable version consistent with the electronic lesson. For signed permissions, offer both written and ESIGN-compliant electronic consent options per district policy.

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