BetterGrades Precalculus · Unit 6 · Lesson

Variation and rational models

Build and interpret direct, inverse, joint, and combined variation models with meaningful parameters and domains.

Opening

Start with the situation

Variation language describes direct, inverse, joint, and combined proportional structure.

Rational graphs are organized around the inputs the denominator forbids. Those exclusions divide the graph into continuity intervals and explain why a simplified expression may still contain a hole or asymptote.

Before you begin

Prerequisite check

Factor numerator and denominator.
Preserve original restrictions.
Use sign and asymptotic notation.

Core explanation

Explanation

Translate the wording into a formula with constant k, use a known data point to find k, and interpret how output responds to input changes.

Inverse models exclude zero and often use positive contextual domains. Inverse variation is not an inverse function.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through variation relationship map, positive-domain inverse model, or another equivalent representation.

Conceptual reading

What the idea is really doing

A rational function carries permanent memory of its original denominator. Factoring may reveal holes, asymptotes, and sign changes, but cancellation never restores an input that the original formula excluded.

This lesson narrows that lens to one goal: build and interpret direct, inverse, joint, and combined variation models with meaningful parameters and domains. 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 the wording into a formula with constant $k$ .
Use a known data point to find $k$ .
Interpret how output responds to input changes.

Verification: Record exclusions first, then compare the factored and simplified forms. Test one point in every sign interval and examine both sides of each vertical asymptote.

Foundation walkthrough

Plan before calculating

Problem

$y$ varies inversely with $x$ ; $y=5$ at $x=8$ .

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Translate the wording into a formula with constant k, use a known data point to find k, and interpret how output responds to input changes.
Conclusion: $y=\frac{40}{x}$
Why the check works: The product xy remains constant.

Worked examples

See the idea in three forms

foundation example

$y$ varies inversely with $x$ ; $y=5$ at $x=8$ .

Solution $y=\frac{40}{x}$

The product xy remains constant.

representation example

Inverse $y=3$ at $x=10$ .

Solution $y=\frac{30}{x}$

This example expresses variation and rational models in a second form.

transfer example

If $x$ doubles in $\frac{k}{x}$ .

Solution $y$ halves.

Inverse models exclude zero and often use positive contextual domains. Inverse variation is not an inverse function.

Variation relationship map. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The product xy remains constant. — **Anchor figure · Variation relationship map**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The product xy remains constant.

Positive-domain inverse model. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for variation and rational models. — **Mechanism figure · Positive-domain inverse model**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for variation and rational models.

Units of k. Compare the valid path with the tempting shortcut. The figure shows why placing a factor in the numerator or denominator contrary to the wording leads to a false conclusion. — **Comparison and error figure · Units of k**
Compare the valid path with the tempting shortcut. The figure shows why placing a factor in the numerator or denominator contrary to the wording leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is placing a factor in the numerator or denominator contrary to the wording.

Check yourself

Direct $y=14$ at $x=2$ .

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

Direct $y=14$ at $x=2$ .

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

Inverse $y=3$ at $x=10$ .

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

If $x$ doubles in $\frac{k}{x}$ .

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

Why $x=0$ excluded?

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: $y=\frac{40}{x}$ . Use the foundation problem as evidence: $y$ varies inversely with $x$ ; $y=5$ at $x=8$ .

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 variation and rational models that it illustrates: Inverse $y=3$ at $x=10$ .

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 placing a factor in the numerator or denominator contrary to the wording.

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: $y$ varies inversely with $x$ ; $y=5$ at $x=8$ . 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: If $x$ doubles in $\frac{k}{x}$ . 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 variation and rational models, 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, Partial-fraction structure, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Lippman and Rasmussen, Precalculus Volume 1
Utah College Algebra
Stitz and Zeager, Precalculus

No long source passage is reproduced.

yyy varies inversely with xxx; y=5y=5y=5 at x=8x=8x=8.

yyy varies inversely with xxx; y=5y=5y=5 at x=8x=8x=8.

Inverse y=3y=3y=3 at x=10x=10x=10.

If xxx doubles in kx\frac{k}{x}xk​.

Original BetterGrades manuscript, rights-separated references.

$y$ varies inversely with $x$ ; $y=5$ at $x=8$ .

$y$ varies inversely with $x$ ; $y=5$ at $x=8$ .

Inverse $y=3$ at $x=10$ .

If $x$ doubles in $\frac{k}{x}$ .