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Course 217 · Life course 39

Read and Build Linear Functions

A linear-model comparison with tables, graphs, slope and intercept interpretations, a solved intersection and a limitation note. Six introductory lessons use supplied practice cases.

Before you start

Use paper or a free text document; every core input is supplied in the lessons.

Solve Equations with Algebra See everything you’ll need

Free download · Original practice data · No external account required. Download the starter pack

What you’ll leave with

A linear-model comparison with tables, graphs, slope and intercept interpretations, a solved intersection and a limitation note.

Start course Start without an account. Sign in to save lesson progress across devices.

Your project

Start with the practice files. Build your project one lesson at a time.

Practice files are free without an account. Sign in to save your project record in My Proof: a description, checklist, and optional work link. Keep the original files in your own app or notebook; saving a project record does not upload them.

  1. Get your materialsDownload practice filesSupplied practice cases, a blank working record, complete lesson instructions, and annotated worked examples.
  2. Make your projectView the project checklist Work in your own notebook, document, or software. Use the supplied practice examples when you prefer.
  3. Keep a private recordSave a project record Sign in to save a description and checklist, then return to update them.
Course handbook

Know the route before you begin.

Here’s what you’ll need and what you’ll make. You can return to this guide during any lesson.

Who this is for

Learners aged 14+ building subject foundations and practical independence. Adults can use the same supplied cases.

Prepare before lesson one
  • Use paper or a free text document; every core input is supplied in the lessons.
  • A basic calculator is optional; show the method, units and checks.
  • Suggested earlier course: Solve Equations with Algebra
Your final project

Finish with proof you can use.

A linear-model comparison with tables, graphs, slope and intercept interpretations, a solved intersection and a limitation note.

Save a project record in My Proof
See project checklist and review criteria

Keep your working files in your own notebook or software. Save a private description, checklist, and work link in My Proof.

What to include

  • Include input/output definitions and bounded domains.
  • Provide checked tables and labeled linear graphs.
  • Interpret rate and starting output with units.
  • Solve an intersection and verify both outputs.
  • Document a capacity-limited model and all changed-case checks.

Review and revise

Use these criteria to check your work and choose what to improve.

  • Method: applies the course-specific rules to the supplied cases with visible intermediate work.
  • Accuracy: results agree with the stated facts, units, boundaries and independent checks.
  • Reasoning: explains why the result follows and identifies a limitation or unresolved question.
  • Transfer: completes changed cases and revises the final project when a relevant assumption changes.
Three ideas to help you along the way
Three useful ideas

The rules behind the steps.

01

Name the quantities before calculating

A function gives exactly one output for each allowed input. The input domain and units matter: a table of a school-day process should not silently be used for negative time or unrelated situations.

02

Use the correct relationship

In y=b+mx, b is the output when x=0 and m is the constant rate. An intercept can be meaningful only if input zero is relevant; it is not automatically a real observed starting condition.

03

Keep the check and boundary visible

A mathematical rule can be evaluated where its real mechanism no longer works. Distinguish calculation from supported prediction and revise the domain when capacity or a changed process invalidates the model.

What do these words mean?
Key words

Key terms worth knowing.

Function
A mapping with exactly one output for each allowed input.
Domain
The permitted input values.
Slope
Output change divided by input change.
Intercept
An axis crossing; the output intercept gives the modeled output at input zero.
Intersection
A point satisfying both model rules.
Extrapolation
Using a model beyond its observed or stated range.
Your route

6 lessons.
One useful project.

Work in order the first time. Each lesson makes something the next lesson can use.

  1. 1
    Up next

    Link inputs, outputs and units

    Define a function’s input, output, units and permitted domain.

  2. 2
    10 min

    Find slope from a table

    Calculate slope from output and input changes across unequal intervals.

  3. 3
    10 min

    Interpret intercepts in context

    Interpret a linear rule’s starting output and rate in context.

  4. 4
    15 min

    Graph a linear relationship

    Plot a linear relationship using labeled numerical axes and an appropriate domain display.

  5. 5
    13 min

    Solve where two models meet

    Solve and verify the shared point of two linear models.

  6. 6
    15 min

    Test a model outside its useful range

    Check observations and identify when a capacity limit invalidates linear extrapolation.

Use the tools

Tools to help you practice.

See all tools
ZIP · practice files

Read and Build Linear Functions practice files

Supplied practice cases, a blank working record, complete lesson instructions, and annotated worked examples.

Text worksheet

Your course project worksheet

A downloadable text worksheet with each lesson’s working output, space for your answers, and the shared final-project checklist. Save your own copy.

Source transparency

Sources and further study.

Explore the references behind the lessons and take your learning further. Source notes explain access and reuse.

OpenStax: linear functions
Source & access notes · checked 2026-10-01

Web open/search showed the official page, its topic heading and relevant concept description. This is a content/source check, not a provider endorsement. Exact topic chapters were opened in web result turn 325view 0–14.

Access: Official page retrieved with the web reader on 2026-10-01; interactive tools and videos were not certified by retrieval.

Reuse: Original fictional teaching cases, calculations, questions and examples; link and short factual paraphrase only. No copied source exercises or illustrations.

OpenStax: modeling with linear functions
Source & access notes · checked 2026-10-01

Web open/search showed the official page, its topic heading and relevant concept description. This is a content/source check, not a provider endorsement.

Access: Official page retrieved with the web reader on 2026-10-01; interactive tools and videos were not certified by retrieval.

Reuse: Original fictional teaching cases, calculations, questions and examples; link and short factual paraphrase only. No copied source exercises or illustrations.

OpenStax: systems by substitution
Source & access notes · checked 2026-10-01

Web open/search showed the official page, its topic heading and relevant concept description. This is a content/source check, not a provider endorsement.

Access: Official page retrieved with the web reader on 2026-10-01; interactive tools and videos were not certified by retrieval.

Reuse: Original fictional teaching cases, calculations, questions and examples; link and short factual paraphrase only. No copied source exercises or illustrations.

Ready when you are

Start with the first useful move.

You’ll always know what to do, what to make, and where to go next.

Start lesson one