BetterGrades Precalculus · Unit 2 · Lesson

Words, tables, graphs, and formulas

Translate a function relationship among verbal, tabular, graphical, and algebraic representations while preserving quantities, units, and domain.

Opening

Start with the situation

Words define quantities and assumptions, tables show selected pairs, graphs reveal global behavior, and formulas support exact calculation.

The same dependency may arrive as a story, table, graph, or formula. Learning to preserve the inputs, outputs, units, and domain while moving among those forms is the central language of the course.

Before you begin

Prerequisite check

Use function notation from the algebra and function readiness unit.
Read ordered pairs and interval notation.
Attach units to contextual quantities.

Core explanation

Explanation

Translate one representation at a time while preserving input, output, units, and domain; then verify agreement at shared values.

Connecting data points claims that intermediate inputs are meaningful, and fitting a formula does not make it an exact law.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through four-representation dashboard, discrete versus continuous connection, or another equivalent representation.

Conceptual reading

What the idea is really doing

A function is a dependency, not merely an equation. Inputs, outputs, units, and domain must agree in words, tables, graphs, and formulas; each representation should tell the same mathematical story.

This lesson narrows that lens to one goal: translate a function relationship among verbal, tabular, graphical, and algebraic representations while preserving quantities, units, and domain. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

Translate one representation at a time while preserving input.
Output.
Units.
Domain; then verify agreement at shared values.

Verification: Choose two representations and make them verify one another. A table can test a formula, a graph can expose a domain or range claim, and units can reveal a model that is algebraically neat but conceptually wrong.

Foundation walkthrough

Plan before calculating

Problem

Values $5,8,11,14$ occur at $x=0,1,2,3$ .

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Translate one representation at a time while preserving input, output, units, and domain; then verify agreement at shared values.
Conclusion: A linear candidate is $f(x)=3x+5$ .
Why the check works: Constant first differences support the candidate.

Worked examples

See the idea in three forms

foundation example

Values $5,8,11,14$ occur at $x=0,1,2,3$ .

SolutionA linear candidate is $f(x)=3x+5$ .

Constant first differences support the candidate.

representation example

Cost $\$25$ plus $\$8$ per person.

Solution $C(p)=25+8p$

This example expresses words, tables, graphs, and formulas in a second form.

transfer example

Best form for exact $f(37)$ .

SolutionFormula.

Connecting data points claims that intermediate inputs are meaningful, and fitting a formula does not make it an exact law.

Four-representation dashboard. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Constant first differences support the candidate. — **Anchor figure · Four-representation dashboard**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Constant first differences support the candidate.

Discrete versus continuous connection. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for words, tables, graphs, and formulas. — **Mechanism figure · Discrete versus continuous connection**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for words, tables, graphs, and formulas.

Exact rule versus fitted model. Compare the valid path with the tempting shortcut. The figure shows why inventing continuity or a unique formula from a small table leads to a false conclusion. — **Comparison and error figure · Exact rule versus fitted model**
Compare the valid path with the tempting shortcut. The figure shows why inventing continuity or a unique formula from a small table leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is inventing continuity or a unique formula from a small table.

Check yourself

Table $f=2x+1$ at $-1,0,1,2$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice

Ten concrete questions

Practice 101

Table $f=2x+1$ at $-1,0,1,2$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 202

Cost $\$25$ plus $\$8$ per person.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 303

Best form for exact $f(37)$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 404

Best form for turning points.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 505

Explain why this conclusion is valid: A linear candidate is $f(x)=3x+5$ . Use the foundation problem as evidence: Values $5,8,11,14$ occur at $x=0,1,2,3$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 606

Solve the representation example, then name the feature of words, tables, graphs, and formulas that it illustrates: Cost $\$25$ plus $\$8$ per person.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is inventing continuity or a unique formula from a small table.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 808

Connect two representations for this example: Values $5,8,11,14$ occur at $x=0,1,2,3$ . Describe what a graph, table, mapping, or algebraic form would have to show.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 909

Create a nearby example by changing one number or condition in this prompt: Best form for exact $f(37)$ . Predict the effect, solve your new example, and compare it with the original.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 1010

Write a short verification checklist for words, tables, graphs, and formulas, then apply it to one worked example from this lesson.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Lesson close

Connect forward

The next lesson, Domain from formulas, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Yoshiwara, Modeling, Functions, and Graphs
Lippman and Rasmussen, Precalculus Volume 1
Stitz and Zeager, Precalculus

No long source passage is reproduced.

Values 5,8,11,145,8,11,145,8,11,14 occur at x=0,1,2,3x=0,1,2,3x=0,1,2,3.

Values 5,8,11,145,8,11,145,8,11,14 occur at x=0,1,2,3x=0,1,2,3x=0,1,2,3.

Cost $25\$25$25 plus $8\$8$8 per person.

Best form for exact f(37)f(37)f(37).

Original BetterGrades manuscript, rights-separated references.

Values $5,8,11,14$ occur at $x=0,1,2,3$ .

Values $5,8,11,14$ occur at $x=0,1,2,3$ .

Cost $\$25$ plus $\$8$ per person.

Best form for exact $f(37)$ .