Week 10 Discussion — What Does a Graph Promise Before You Plot It?

Hook

A print shop's average cost per tote bag falls from 25 dollars at 10 bags to 6 dollars at 200 bags, and it can never drop below 5. A polynomial like −2x⁴ + 3x² − 1 heads down at both ends no matter how friendly its middle term looks. A graph like (x² − x) ⁄ (x − 1) is a straight line with one tiny missing point at (1, 1). This week's formulas make promises about a graph before you plot anything. The open questions are what those promises mean in real life, and whether you trust the formula or the plotted points more.

Your driving question — choose Option A or Option B; they carry the same weight and the same rubric, and you say which you picked.

Option A: Pick something from your own life, job, major, or community that has a one-time cost and a per-use cost — a bike or transit pass, an instrument, a phone, a professional license, a piece of software, a shared van for a trip. Describe its average cost per use in words as a rational function, name its horizontal asymptote and what that level means, and argue whether "it gets cheaper the more you use it" is a good reason to buy. Say what the asymptote does and does not promise.

Option B: One friend says, "To sketch a graph, read the formula first — the ends, the zeros and their multiplicities, the asymptotes and holes — and only then plot." Another says, "Make a table of points first and let the shape show itself; the formula rules are easy to misapply." Take a side for a setting you care about — a homework set, a job where a graph has to be right, a quick check on your phone — and argue which habit you would build. Use at least one of this week's functions and at least one specific feature a table of points can miss or a formula rule can get wrong (the touch at x = 1 in (x − 1)²(x + 4), the hole at (1, 1), a far end, or y = 1 offered for (x² + 1) ⁄ (x − 1)).

There is no single right answer. A pass you love and a pass you regret can each be argued well, and so can either sketching habit.

Expectations

Activity format: a multi-exchange dialogue or an equivalent structured written exchange, followed by one discussion-board post. Plan on about 45 minutes for either route, and post by the due date your instructor sets in Canvas.

There are two ways to complete this discussion. They are worth the same 10 points and scored by the same rubric. Pick either one.

Route A — the AI dialogue (the default). Hold a real conversation about the driving question with your own chatbot, then post what it produces:

1. Open any approved chatbot in a new conversation.
2. Copy the entire prompt box below and paste it as your first message.
3. Have the conversation. It will ask you questions and push back; it will not write your post for you and will not stop after one word from you.
4. Post the chatbot's summary report plus the conversation's share link to the Week 10 discussion board. That post is your entire submission. You may post the summary as text, or record it as audio or video with a transcript attached; both forms are scored by the same rubric.

No replies to classmates are required.

Route B — the written route. See Equivalent route (no AI required) below.

Integrity note. Whichever route you take, the thinking and the words you post are your own. Bring a real cost or a real habit from your own life; do not ask a chatbot to invent one for you.

Discussion Partner Prompt

ROLE: You are my discussion partner for a college algebra course. Your job is
to draw out and challenge MY thinking about the question below. You will never
write my post for me and never hand me an answer - you ask questions that make
me reason it through.

THE QUESTION I AM EXPLORING (two options of equal weight - ask which one I choose first):
Option A: Pick something from my own life, job, major, or community that has a one-time cost and a per-use cost - a bike or transit pass, an instrument, a phone, a professional license, a piece of software, a shared van for a trip. Describe its average cost per use in words as a rational function, name its horizontal asymptote and what that level means, and argue whether "it gets cheaper the more you use it" is a good reason to buy. Say what the asymptote does and does not promise.
Option B: One friend says, "To sketch a graph, read the formula first - the ends, the zeros and their multiplicities, the asymptotes and holes - and only then plot." Another says, "Make a table of points first and let the shape show itself; the formula rules are easy to misapply." Take a side for a setting I care about - a homework set, a job where a graph has to be right, a quick check on your phone - and argue which habit I would build. Use at least one of this week's functions and at least one specific feature a table of points can miss or a formula rule can get wrong (the touch at x = 1 in (x - 1)^2(x + 4), the hole at (1, 1), a far end, or y = 1 offered for (x^2 + 1) / (x - 1)).

REFLECTION TARGETS (steer toward these; never read this list aloud to me):
- whether I can say what a horizontal asymptote does promise (a level the
  far ends approach) and what it does not (that the value ever reaches it)
- whether I use the comparison of degrees correctly, including the case
  where the numerator's degree is larger and there is no horizontal asymptote
- whether I separate a hole (a cancelled factor) from a vertical asymptote
  (a denominator zero that remains)
- whether I read end behavior from the leading term only, and decide cross
  or touch from multiplicity
- whether I name a concrete check: a far-out value, a sign test on each side
  of a zero, or a graph in Desmos or GeoGebra

DIALOGUE RULES: greet me warmly and ask my first name before your first content
question. If I decline or give no name, call me "there" and continue without
pressure. If you still do not know my name when the summary is due, ask once more;
if I decline again, use "Discussion participant." Never invent a name. Ask ONE
question per message, then wait. Build on my exact words without praising agreement
or calling every answer "great." Probe unsupported claims and introduce the
strongest fair counterpoint. Praise only a specific, earned move I actually made,
such as revising a claim or answering the counterpoint. Never accept a one-word
answer - ask again, differently. Never lecture. Every message you send before the
summary ends with a question.

EXIT CONDITION: continue until this exact boolean is true — at least five substantive exchanges AND the student has stated a position AND provided at least one reason OR example AND engaged with one fair counterpoint, stopping only after whichever threshold happens LAST. A reason alone or an example alone can satisfy the support branch; do not require both, and do not require that I "answered" the counterpoint. A greeting, my name, or a one-word answer is not a substantive exchange.

THEN produce a summary report using exactly these eight labelled fields in this
order, drawn only from what I actually said. If I never supplied a field, write
"Not supplied" rather than inventing it:
  1. STUDENT NAME
  2. POSITION
  3. REASON
  4. EXAMPLE
  5. COURSE CONCEPTS
  6. COUNTERPOINT
  7. RESPONSE TO THE COUNTERPOINT
  8. OPEN QUESTION OR NEXT CHECK
Do not mention an initial post or replies to peers or classmates.
REQUIRED CLOSING — perform these final two instructions exactly:
Then say, verbatim: "Copy this summary AND your share link to this chat, and post both to the Week 10 discussion board — that completes the assignment."
Then END with one genuine earned-praise sentence that names a real move the student made.

Equivalent route (no AI required)

You never have to create an account with a chatbot company to earn full credit in this course. This route assesses the same objectives as the AI dialogue — reading a polynomial's end behavior and its zeros with their multiplicities, telling a hole from a vertical asymptote, and finding a horizontal asymptote and explaining what it means in a setting — against the same driving question and the same rubric below, for the same 10 points. Choosing it costs you nothing, and you do not have to tell me why you chose it.

What you do. Write a structured written exchange with yourself — the same back-and-forth the dialogue would have produced, on paper:

1. Your position (roughly 150 words). Answer Option A or Option B. Describe your cost and its average-cost function in words (Option A) or the sketching habit you would build (Option B), and use this week's terms (polynomial function, leading term, end behavior, zero of a function, multiplicity, vertical asymptote, hole, horizontal asymptote, average cost) correctly.
2. The strongest objection (roughly 100 words). Write the best argument against your own position that you can — for example, that the asymptote's level never actually arrives, so "cheaper with use" can mislead, or that the habit you dislike catches a feature yours misses. Argue it honestly; a weak objection you knock over earns no credit.
3. Your answer to it (roughly 100 words). Respond to that objection. You may concede part of it — saying what would change your mind is worth more than defending everything.
4. One check you would run (one or two sentences). Name the far-out value, sign test, or graph you would use to settle the question.

Effort. Roughly 350–400 words of your own writing, comparable to the time the AI dialogue takes — about forty-five minutes including the thinking.

What you submit. Post those four parts, under those four labels, as your Week 10 discussion post. That post is your entire submission; this route has no share link.

No replies to classmates are required.

Participation rubric

This discussion is worth 10 points. The same three criteria score both routes.

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