Week 10 Discussion — What Does a Graph Promise Before You Plot It? — Traditional alternative (no AI)

Instructors: publish this instead of the adaptive version if you prefer, and unpublish or delete the adaptive one.

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 discussion 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 here. A pass you love and a pass you regret can each be argued well, and so can either sketching habit. Pick one you actually care about, and tell us the story.

Expectations

Activity format: one initial post followed by two substantive peer replies. Plan on about 45 minutes across the two deadlines; your instructor sets both due dates in Canvas.

Your initial post — by the first due date (early in the module week):

• Say which option you chose.
• For Option A, describe the cost, its average-cost function in words, its horizontal asymptote, and what that level does and does not promise. For Option B, name your habit and the specific feature it catches or misses.
• Use at least two of this week's terms (polynomial function, leading term, end behavior, zero of a function, multiplicity, vertical asymptote, hole, horizontal asymptote, average cost) correctly.
• Give your judgment with a reason, and name one concrete check: a far-out value, a sign test on each side of a zero, or a graph.
• Length: roughly 150–250 words if written. You may instead record your post as audio or video (about two minutes) and attach a transcript; both forms are scored by the same rubric.

Your replies — by the second due date (the end of the module week):

• Reply substantively to at least two classmates.
• A substantive reply does more than agree: test a classmate's asymptote by checking a far-out value, ask whether their excluded input is a hole or an asymptote, or point out a feature their favorite sketching habit could miss. "Nice point!" on its own does not count.

What a strong post looks like:

I chose Option A. A bike-share pass costs 90 dollars for the season, and each ride adds 1 dollar, so my average cost per ride is (x + 90) ⁄ x for x rides. At 30 rides that is 120 ⁄ 30 = 4 dollars a ride. The degrees are equal, so the horizontal asymptote is y = 1: the average falls toward 1 dollar but never gets there, because the 90 dollars is always shared. "Cheaper with use" is true, but the drop slows down, so the pass only pays off if I ride often enough early.

Notice how it builds the function from the real costs, checks one value, says what the asymptote does and does not promise, and ties it to a decision. That is the bar.

Participation rubric

This discussion is worth 10 points. The three criteria below look at your initial post and your peer replies.

Post early if you can — the best threads happen when people have time to reply before the second due date.

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