Week 10 — Lecture Outline · Polynomial & Rational Functions
Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objectives covered: Objective 7 — Analyze polynomial and rational functions using end behavior, zeros and multiplicity, domain, and vertical and horizontal asymptotes.
SLOs touched: A (apply procedures accurately) · B (connect symbolic/graphical representations and interpret in context)
Meeting pattern: 2 sessions × 75 min = 150 min. Segment minutes below total ~150; scale to your own pattern.
Week at a Glance
| The week's big question | "Without plotting a single point, what can the leading term, the zeros, and the asymptotes tell you about the entire shape of a function's graph?" |
| By the end of the week, students can… | (1) determine end behavior of any polynomial from its leading term (even/odd degree + sign of leading coefficient); (2) identify zeros and multiplicity in factored form and state whether the graph crosses (odd) or touches (even) at each; (3) state the domain of a rational function and identify vertical asymptotes vs. holes; (4) find the horizontal asymptote by comparing degrees of numerator and denominator. |
| Key vocabulary | end behavior, leading term, degree, even/odd degree, leading coefficient, zero, multiplicity (even/odd), rational function, domain, excluded value, vertical asymptote, hole, horizontal asymptote |
| Materials | slides (Deck 10), the week's readings + video links, Desmos for graphing checks, one approved chatbot (Gemini / Claude / ChatGPT) for the AI-critique moment and the tutorial |
| Timing note | 8 segments, ~150 min total. Session 1 = Segments 1–4 (~73 min). Session 2 = Segments 5–8 (~77 min). |
Segment 1 — Hook & the Promise (8 min) · Session 1 opens
Hook. "Has anyone ever seen an average-cost curve — say, the cost per unit to produce T-shirts when you share the fixed setup costs across more and more units?" Draw a rough curve: drops steeply, then flattens and approaches a floor it never quite reaches.
- "That floor is a horizontal asymptote — a value the function gets infinitely close to but never hits. The curve is the graph of a rational function, and today we learn how to read everything about that shape from the algebra alone, without plotting a single point."
- "Same idea comes up in pharmacology — drug concentration in your bloodstream approaches zero as your body processes it, but it never actually reaches zero. In physics, a falling object approaches terminal velocity. Everywhere you have a process that's bounded — a limit you approach but never reach — you'll find an asymptote."
The promise (write it on the board): "By the end of this week you can look at a polynomial's leading term and immediately tell me which way both tails point — and look at a rational function and tell me its domain, its asymptotes, and the shape of its graph, all from the algebra."
Why it matters line (memory hook): "End behavior, zeros, and asymptotes are the skeleton of a function's graph. Get the skeleton right and you never need to plot 50 points again."
Segment 2 — Polynomial End Behavior (20 min)
Plain language first. When x gets very large or very small (far left, far right of the number line), the graph of a polynomial is dominated by its leading term — the term with the highest power. Everything else becomes negligible compared to the leading term when x is huge.
The two decisions that determine end behavior:
1. Degree — odd or even?
- Even degree (2, 4, 6, …): both tails behave the same way — both up or both down.
- Odd degree (1, 3, 5, …): the tails go in opposite directions — one up and one down.
2. Leading coefficient — positive or negative?
- Positive: the right tail (x → +∞) goes up.
- Negative: the right tail goes down.
Memory hook: "Even degree = same-direction tails; odd degree = opposite tails. Positive leading coefficient = right tail up; negative = right tail down."
Four cases (put on a 2×2 grid on the board):
| Degree | Leading coeff | Left tail | Right tail | Common example |
|--------|---------------|-----------|------------|----------------|
| Even | Positive | Up (↑) | Up (↑) | x², x⁴ |
| Even | Negative | Down (↓) | Down (↓) | −x², −2x⁴ |
| Odd | Positive | Down (↓) | Up (↑) | x, x³ |
| Odd | Negative | Up (↑) | Down (↓) | −x, −x³ |
One fully worked example — every step out loud:
Describe the end behavior of f(x) = −2x⁴ + 3x² − 1.
1. Leading term: −2x⁴ (ignore the rest for end behavior).
2. Degree: 4 → even → both tails go the same way.
3. Leading coefficient: −2 → negative → tails go down.
4. Conclusion: as x → ±∞, f(x) → −∞. The graph falls on both ends.
(Common error: using the −1 or 3 instead of the degree-4 term. The leading term always wins for large x.)
One more worked example:
Describe the end behavior of g(x) = x³ − 5x + 2.
1. Leading term: x³.
2. Degree: 3 → odd → tails go in opposite directions.
3. Leading coefficient: +1 → positive → right tail up.
4. Conclusion: as x → +∞, g(x) → +∞; as x → −∞, g(x) → −∞. The graph falls left, rises right.
Segment 3 — Zeros and Multiplicity (22 min)
Plain language first. We've always found zeros by setting f(x) = 0 and solving. Now we care about how many times each factor is repeated — the multiplicity — because it tells us the graph's behavior at that zero.
The rule (state it plainly, then give the reason):
- Even multiplicity (2, 4, …): the graph touches the x-axis at the zero and turns back (bounces). Think of x² = 0 at x = 0: the parabola just kisses the axis.
- Odd multiplicity (1, 3, …): the graph crosses the x-axis at the zero. Think of x³ = 0 at x = 0: it punches through.
Memory hook: "Even multiplicity = even behavior (gentle, turns back). Odd multiplicity = odd behavior (crosses, keeps going)."
One fully worked example — every step:
Find the zeros of f(x) = (x − 1)²(x + 4) and state what the graph does at each.
1. Set each factor to zero: x − 1 = 0 → x = 1; x + 4 = 0 → x = −4.
2. Check multiplicities: (x − 1)² → multiplicity 2 (even) → graph touches x-axis at x = 1 and turns back.
3. (x + 4)¹ → multiplicity 1 (odd) → graph crosses x-axis at x = −4.
4. Summary: two zeros — at x = 1 (touches) and x = −4 (crosses).
One more (three factors):
Find zeros and behavior for f(x) = x²(x − 3)(x + 2)³.
- x = 0: multiplicity 2 (even) → touches.
- x = 3: multiplicity 1 (odd) → crosses.
- x = −2: multiplicity 3 (odd) → crosses.
Name the misconception out loud:
- ❌ "A zero with multiplicity 2 must cross the x-axis — after all, it's a double root."
✅ Cure: multiplicity even means the graph never crosses — it touches and reverses direction. Think of the parabola y = (x − 1)²: it reaches y = 0 at x = 1 and bounces back up; it never goes negative.
Quick interaction (~4 min): Give students a polynomial in factored form; each person writes down (a) the zeros, (b) the multiplicity of each, and (c) "cross" or "touch." Compare with a neighbor, then call on volunteers to share.
Segment 4 — Misconceptions on End Behavior + Multiplicity (10 min) · Session 1 closes (~73)
Name and cure the classic errors systematically:
-
❌ "f(x) = x³ rises on both sides — it's a cubic and big cubed numbers are positive."
✅ Cure: at x = −10, x³ = −1000 → very negative. Odd degree means opposite tails. Only even degree can be same-direction. -
❌ "For f(x) = −x², the negative makes the exponent act odd."
✅ Cure: the degree determines odd/even, not the sign. Degree 2 is always even, so both tails behave the same. The negative sign just flips which direction they go (down instead of up). -
❌ "A zero with even multiplicity crosses the x-axis twice."
✅ Cure: at a zero of even multiplicity the graph does not cross — it touches and turns back. It only "returns to" zero once, right at that point.
Instructor note: spend a minute on the end-behavior 2×2 grid from Segment 2 — ask students to reproduce it from memory. This is the most-tested idea on the quiz.
Segment 5 — Rational Functions: Domain & Vertical Asymptotes (22 min) · Session 2 opens
Hook back in: "Last session: polynomials have predictable end behavior and zeros. Today: divide one polynomial by another and something new appears — values of x where the function simply doesn't exist."
Plain language first — domain of a rational function:
A rational function is a ratio of two polynomials: f(x) = P(x) / Q(x). Division by zero is undefined, so the domain of f excludes all values of x where Q(x) = 0.
Worked example (domain):
Find the domain of f(x) = (x + 1) / (x² − 4).
1. Factor the denominator: x² − 4 = (x − 2)(x + 2).
2. Set each factor to zero: x = 2, x = −2.
3. Domain: all real numbers except x = 2 and x = −2, written x ≠ 2, x ≠ −2 (or in interval notation: (−∞, −2) ∪ (−2, 2) ∪ (2, +∞)).
Vertical asymptotes vs. holes:
Once you have the excluded values, check whether the numerator also equals zero at those x-values:
- If Q(a) = 0 and P(a) ≠ 0: the function blows up → vertical asymptote at x = a.
- If Q(a) = 0 and P(a) = 0 (the factor cancels): the function has a hole (removable discontinuity), not a vertical asymptote.
Worked example (vertical asymptotes — no cancellation):
Find the vertical asymptotes of f(x) = (x + 1) / (x² − 4).
Excluded values: x = 2 (numerator: 2 + 1 = 3 ≠ 0 → vertical asymptote) and x = −2 (numerator: −2 + 1 = −1 ≠ 0 → vertical asymptote).
Result: vertical asymptotes at x = 2 and x = −2.
Worked example (hole vs. asymptote — with cancellation):
Analyze g(x) = (x² − x) / (x − 1).
1. Factor numerator: x(x − 1).
2. g(x) = x(x − 1)/(x − 1). The factor (x − 1) cancels.
3. At x = 1: Q(1) = 0, but the factor cancels → hole at x = 1, not a vertical asymptote.
4. The simplified form is g(x) = x (x ≠ 1) — a line with a hole punched out at (1, 1).
Named misconception:
- ❌ "Every denominator zero is a vertical asymptote."
✅ Cure: if the factor also cancels in the numerator, it's a hole, not an asymptote. Always simplify first.
Segment 6 — Horizontal Asymptotes (20 min)
Plain language first. A horizontal asymptote describes what happens to a rational function's output as x → ±∞ (far out to the right and left). Unlike vertical asymptotes (where the function blows up), the function doesn't blow up here — it approaches a fixed y-value from above or below.
The three cases — compare degrees of numerator and denominator:
| Degrees | Horizontal asymptote | Plain-language reason |
|---|---|---|
| deg(P) < deg(Q) | y = 0 | Denominator grows faster; the fraction shrinks to zero |
| deg(P) = deg(Q) | y = (leading coeff of P) / (leading coeff of Q) | The ratio of the dominant terms takes over |
| deg(P) > deg(Q) | None | Numerator grows faster; the fraction blows up |
Memory hook: "Less-than → floor at zero. Equal → the leading-coefficient ratio. Greater-than → no ceiling, no floor — it heads to infinity."
Three worked examples (one per case):
(Case 1: deg numerator < deg denominator)
Find the horizontal asymptote of h(x) = 2x / (x² + 1).
- deg(2x) = 1; deg(x² + 1) = 2. Since 1 < 2: y = 0.(Case 2: equal degrees)
Find the horizontal asymptote of f(x) = (3x + 1) / (x − 5).
- deg(3x + 1) = 1 = deg(x − 5) = 1. Leading coefficients: 3 (numerator) and 1 (denominator). y = 3/1 = 3.(Case 3: deg numerator > deg denominator)
Find the horizontal asymptote of p(x) = (x² + 1) / (x − 1).
- deg(x² + 1) = 2 > deg(x − 1) = 1. No horizontal asymptote. (The function grows without bound as x → ±∞.)
Named misconception:
- ❌ "If both numerator and denominator have an x², the horizontal asymptote is y = 1."
✅ Cure: when degrees are equal, the asymptote is the ratio of the leading coefficients, not just 1. For (3x²)/(2x²), the asymptote is y = 3/2, not y = 1. Always read the coefficients.
Segment 7 — Putting It Together: Sketch the Key Features (15 min)
Plain language first. The goal isn't to plot 50 points — it's to use end behavior + zeros + asymptotes as a skeleton, then sketch a smooth curve through those constraints.
Framework — six-step sketch (give students a handout or put on board):
1. Factor numerator and denominator completely.
2. Zeros of the denominator → excluded values → check each: vertical asymptote or hole.
3. Degree comparison → horizontal asymptote (if any).
4. Zeros of the numerator → x-intercepts; determine cross vs. touch via multiplicity.
5. y-intercept → set x = 0 and evaluate.
6. End behavior for polynomials (leading term), or asymptotes for rational functions; sketch the tails.
One fully worked synthesis (rational function):
Sketch the key features of f(x) = (x + 1) / (x² − 4).
1. Factor: (x + 1) / [(x − 2)(x + 2)]. No cancellation.
2. Vertical asymptotes: x = 2 and x = −2.
3. Horizontal asymptote: deg(1) < deg(2) → y = 0.
4. x-intercept: x + 1 = 0 → x = −1 (crosses; multiplicity 1).
5. y-intercept: f(0) = 1/(0 − 4) = −1/4.
6. Sketch: two vertical dashed lines at x = ±2; horizontal dashed line at y = 0; curve passes through (−1, 0) and (0, −1/4); tails approach y = 0 on both far ends.
Segment 8 — Technology Workflow + AI-Critique, Callback & Hand-off (13 min) · Session 2 closes (~77)
Technology workflow — verify asymptotes in Desmos (exact steps):
1. Open desmos.com/calculator (free, no login).
2. Type the rational function on line 1: (x+1)/(x^2-4).
3. Type the horizontal asymptote on line 2: y = 0.
4. Type vertical asymptotes on lines 3–4: x = 2 and x = -2.
5. Observe: the graph approaches but never touches the asymptotes. Zoom out (pinch or scroll) to see both tails level off toward y = 0.
6. Same trick for any rational function: graph it + its asymptotes; the picture confirms or corrects your algebra.
AI-critique moment (students verify, not consume):
Paste this to an approved chatbot: "Find the horizontal asymptote of f(x) = (x² + 3) / (x − 2) and explain."
The correct answer: deg numerator (2) > deg denominator (1) → no horizontal asymptote. Common chatbot error: the model says "y = x + 2" (the oblique asymptote from polynomial long division) — which is beyond our scope — OR it says "y = 1" thinking the leading coefficients are both 1 but ignoring the degree mismatch. Either way, it gets the HA question wrong for our course's purposes. Your job all semester: the tool drafts, you judge. This is exactly how the weekly Lecture Tutorial works — you'll catch the model, not trust it.
Callback + tease:
- Callback: "Factoring from Week 6 is the key that unlocks this week — you can't find zeros or cancel factors without it. And the parabola work from Week 9 is a special case of today's polynomial analysis."
- Tease next week: "Week 11 takes rational functions one step further: adding, subtracting, multiplying, and dividing them algebraically, and — importantly — solving rational equations, where extraneous solutions lurk."
Hand-off (the week's graded work):
- Lecture Tutorial 10 (AI tutor, share-link submission) — end behavior, zeros/multiplicity, domain, vertical asymptotes, horizontal asymptotes.
- Quiz 10 (end of week, no AI) and Discussion 10 ("Asymptotes in the Wild").
- Assignment 10 ("Shape Without Plotting") — AI-coached, self-scored.
Instructor FAQ — Common Stumbles
| Student says / does | Quick cure |
|---|---|
| "f(x) = −x³ falls on both sides — negative." | Degree 3 is odd, so tails go in opposite directions. The negative sign only flips which side is up — left tail rises, right tail falls. |
| Gives end behavior based on the sign of a middle term. | End behavior depends only on the leading term (highest-degree term). All other terms are irrelevant as x → ±∞. |
| "x = 1 is a vertical asymptote in (x − 1)²/(x − 1)." | (x − 1) cancels → hole at x = 1, not a vertical asymptote. Always simplify first. |
| Says horizontal asymptote of (3x + 1)/(x − 5) is y = 1 (equal degrees → 1). | Equal degrees → ratio of leading coefficients = 3/1 = 3. Read the coefficients, not just the degree. |
| Says horizontal asymptote of (x² + 1)/(x − 1) is y = x (or y = x + 1). | Degree of numerator is greater → no horizontal asymptote (there's an oblique one, but that's beyond our scope this week). |
| "Even multiplicity means the zero is crossed twice." | Even multiplicity means the graph touches but does not cross. Think of y = x² at x = 0: it never goes below the x-axis. |
| "The domain restriction and the vertical asymptote are the same thing." | A domain restriction is any x-value excluded (denominator = 0). A VA is a domain restriction where the factor doesn't cancel. A hole is a domain restriction where it does cancel. |
Scope flag
This outline stays within Objective 7 (end behavior, zeros/multiplicity, domain, vertical and horizontal asymptotes). Oblique (slant) asymptotes, polynomial long division for the "numerator-degree-higher" case, and the full limit-based treatment of asymptotes are out of scope — mentioned only to name what the chatbot sometimes suggests, not to teach it. Formal limits are a Calculus I concept.
~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com