Study Guide
Learning Objectives
- Determine holes in graphs of rational functions.
- Apply: num multiplicity ≥ den multiplicity at shared real zero $c$ ⇒ hole at $x=c$.
- Locate the hole as $(c,L)$ where $\lim_{x\to c}r(x)=L$.
- Distinguish holes from vertical asymptotes (1.9) and zeros (1.8).
Suggested Practice Sequence
1. Equal-multiplicity holes; find $(c,L)$. 2. Num mult > den mult holes. 3. Contrast items that are VAs instead. 4. Limit language both sides agree. 5. Graphs: open circle at $(c,L)$. 6. Mixed FRQs sorting hole / VA / zero / HA.
Classroom / Check-In Moves
- Require multiplicity table before labeling.
- Cold-call: “What’s $L$?” after every hole $x=c$.
- Sort cards: hole vs VA using only multiplicity info.
Misconception Watch-List
- Shared factor ⇒ always hole.
- Hole ⇒ $r(c)$ is defined after canceling on paper.
- Hole $y$-coordinate always 0.
Vocabulary to Own
- hole / removable discontinuity
- multiplicity comparison
- $\lim_{x\to c}r(x)=L$
- simplified rational expression
- open circle on a graph
Materials Map
- Practice: classify + locate holes
- Word: removable gap stories
- Mixed / multi-*: contrast with VA/zeros/HA
- Review & practice test: timed hole fluency
- Interactive: quick checks