BetterGrades Precalculus · Unit 6 · Lesson

Intercepts and sign charts

Find rational intercepts and determine positive and negative intervals using factored sign structure.

Opening

Start with the situation

A rational function has constant sign between numerator zeros and denominator zeros.

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

Factor, classify zeros as intercepts, holes, or asymptotes, test one input per interval, and calculate the y-intercept if allowed.

Even multiplicity preserves sign; odd multiplicity changes sign.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through rational sign chart, intercept versus hole, 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: find rational intercepts and determine positive and negative intervals using factored sign structure. 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

Factor.
Classify zeros as intercepts.
Holes.
Or asymptotes.

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

Analyze signs of $\frac{x-1}{x+2}$ .

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Factor, classify zeros as intercepts, holes, or asymptotes, test one input per interval, and calculate the y-intercept if allowed.
Conclusion: Positive, negative, positive across $-2$ and $1$ .
Why the check works: Critical values split the sign intervals.

Worked examples

See the idea in three forms

foundation example

Analyze signs of $\frac{x-1}{x+2}$ .

SolutionPositive, negative, positive across $-2$ and $1$ .

Critical values split the sign intervals.

representation example

Y-intercept.

Solution $(0,-5)$

This example expresses intercepts and sign charts in a second form.

transfer example

Critical values of $\frac{x}{(x-2)(x+4)}$ .

Solution $-4,0,2$

Even multiplicity preserves sign; odd multiplicity changes sign.

Rational sign chart. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Critical values split the sign intervals. — **Anchor figure · Rational sign chart**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Critical values split the sign intervals.

Intercept versus hole. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for intercepts and sign charts. — **Mechanism figure · Intercept versus hole**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for intercepts and sign charts.

Multiplicity at rational zero. Compare the valid path with the tempting shortcut. The figure shows why counting a cancelled numerator zero as an intercept leads to a false conclusion. — **Comparison and error figure · Multiplicity at rational zero**
Compare the valid path with the tempting shortcut. The figure shows why counting a cancelled numerator zero as an intercept leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is counting a cancelled numerator zero as an intercept.

Check yourself

X-intercept of $\frac{x-5}{x+1}$ .

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Attempt once to unlock the answer

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

Practice

Ten concrete questions

Practice 101

X-intercept of $\frac{x-5}{x+1}$ .

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

Y-intercept.

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Attempt once to unlock the answer

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Practice 303

Critical values of $\frac{x}{(x-2)(x+4)}$ .

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

Can a hole be intercept?

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Practice 505

Explain why this conclusion is valid: Positive, negative, positive across $-2$ and $1$ . Use the foundation problem as evidence: Analyze signs of $\frac{x-1}{x+2}$ .

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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 intercepts and sign charts that it illustrates: Y-intercept.

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Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is counting a cancelled numerator zero as an intercept.

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Practice 808

Connect two representations for this example: Analyze signs of $\frac{x-1}{x+2}$ . Describe what a graph, table, mapping, or algebraic form would have to show.

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Practice 909

Create a nearby example by changing one number or condition in this prompt: Critical values of $\frac{x}{(x-2)(x+4)}$ . Predict the effect, solve your new example, and compare it with the original.

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Practice 1010

Write a short verification checklist for intercepts and sign charts, then apply it to one worked example from this lesson.

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Lesson close

Connect forward

The next lesson, Complete rational graph construction, 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.

Analyze signs of x−1x+2\frac{x-1}{x+2}x+2x−1​.

Analyze signs of x−1x+2\frac{x-1}{x+2}x+2x−1​.

Y-intercept.

Critical values of x(x−2)(x+4)\frac{x}{(x-2)(x+4)}(x−2)(x+4)x​.

Original BetterGrades manuscript, rights-separated references.

Analyze signs of $\frac{x-1}{x+2}$ .

Analyze signs of $\frac{x-1}{x+2}$ .

Critical values of $\frac{x}{(x-2)(x+4)}$ .