Study Guide

Learning Objectives

• Determine real zeros of rational functions (numerator zeros in the domain).
• Exclude denominator zeros from the zero list.
• Use real zeros of both numerator and denominator as endpoints/asymptotes for $r(x)\ge 0$ or $r(x)\le 0$ intervals.
• Distinguish zeros of $r$ from domain restrictions (VA/holes preview).

Suggested Practice Sequence

1. Simple $r$ with distinct num/den zeros. 2. Cases where a numerator root is excluded. 3. Shared-factor previews (not a zero of $r$). 4. Sign charts / inequality intervals. 5. Graph language: intercepts vs breaks. 6. Mixed multi-step with 1.7 end behavior (zeros ≠ ends).

Classroom / Check-In Moves

• Require domain statement before zero list.
• Cold-call: “Is this numerator root in the domain?”
• Inequality warm-up: mark all critical values first.

Misconception Watch-List

• Denominator zero ⇒ function zero.
• Inequality critical set = numerator only.
• Canceled factor still an $x$-intercept.

Vocabulary to Own

• zero of a rational function
• domain restriction
• $x$-intercept
• critical values / test intervals
• rational inequality

Materials Map

• Practice: find zeros with domain filter
• Word: applied break-even / sign stories
• Mixed / multi-*: inequalities + links to 1.7/1.9
• Review & practice test: timed zero/inequality fluency
• Interactive: quick checks