Week 9 — Finding the Turning Point: A Short Parabola Report

Assignment 8 of the term — work four short problems about parabolas, explain what each result means, and tell someone new to algebra why the best garden sits at a turning point.

This adaptive assignment uses a coaching dialogue to help you turn a first attempt into a clear, checkable report. You remain the author. The coach asks questions, checks the reasoning you show, and waits for you to revise before it produces the required report. In Week 7 you solved x² − 6x + 5 = 0 and found its two answers, 1 and 5; this week you read the same expression as a function with a vertex, an axis of symmetry, and intercepts, find maximum and minimum values, and switch between vertex form and standard form. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles; this assignment is where you put them together and explain them.

The task

The week's parabola is f(x) = x² − 6x + 5, whose outputs for x = 0 through 6 are 5, 0, −3, −4, −3, 0, and 5. You will also use the parabola f(x) = −x² + 4x + 1, and the launched object whose height in feet t seconds after launch is h(t) = −16t² + 64t + 80.

Work these four problems, showing your arithmetic:

1. The week's parabola f(x) = x² − 6x + 5 has a = 1 and b = −6. Using the vertex formula h = −b ⁄ (2a), what is the x-value h of its vertex?
2. The parabola f(x) = −x² + 4x + 1 has its vertex at x = 2. What is its maximum value k = f(2)?
3. A launched object's height in feet t seconds after launch is h(t) = −16t² + 64t + 80. How many feet high is its peak?
4. Complete the square to rewrite the week's parabola x² − 6x + 5 in vertex form (x − 3)² + k, using half of b, which is −3. What is k?

Beside each answer, add one brief interpretation: what that value shows about the parabola and one use or limit of it. For problem 1, say what h tells you about the graph, give the full vertex, and say whether it is a minimum or a maximum. For problem 2, say why the value is a maximum and state the range. For problem 3, say when the peak happens as well as how high it is, and name one limit of the model. For problem 4, write the full vertex form, say what k means and how it matches the vertex from problem 1, and check both forms at x = 0.

Explain it to a non-expert

A neighbor is planning a rectangular vegetable garden and has exactly 40 feet of fencing. They have never studied algebra, and they ask you which shape holds the most. The area for a width of w feet is A(w) = w(20 − w) square feet. Write a 3–5 sentence message your neighbor could follow: why the best width sits at the turning point of that area, what the best width and the largest area are in everyday words, and one honest limit of the answer, such as a yard that is too narrow for that shape. Explain any technical term in everyday words. More than one good limit can be defended.

For a different, ungraded example, imagine a band selling T-shirts at a show. If it charges p dollars a shirt, it expects to sell 30 − p shirts, so the money it takes in is R(p) = p(30 − p) dollars. Model answer: "If the shirts are cheap, lots of fans buy them, but each sale brings in very little. If they are expensive, each sale brings in a lot, but few fans buy. The money climbs and then falls, and it is greatest right at the turning point, 15 dollars a shirt, which brings in 225 dollars. The model assumes every fan reacts to price the same way, so treat 15 dollars as a starting guess to try, not a promise."

Use that pattern for the garden, with your own explanation.

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message for the neighbor before opening a chatbot.
2. Open any course-approved chatbot. Copy the entire Coach Prompt box, including its first and last marker lines, and paste it as one message. The coach will ask for your draft and then work one question at a time.
3. Answer in your own words. Show calculations when asked. Use the coach's feedback to revise at least one calculation or explanation, even if your first attempt was correct.
4. Continue until the prompt's exit condition is met.
5. Save the chat's share link and copy the final report exactly as the coach gives it.

The chatbot is a coach, not a ghostwriter. It must draw out your reasoning rather than write the report for you.

Coach Prompt

COPY EVERYTHING BELOW THIS LINE
ROLE: You are a careful learning coach. Draw out and challenge MY thinking about quadratic functions: the vertex, the axis of symmetry, minimum and maximum values, the vertex formula, intercepts, and converting between vertex form and standard form. Do not write my assignment for me and do not invent work I did not show.

ASSIGNMENT: Work through these four problems in order, one at a time. Do not invent, replace, or reorder a problem. The data: the week's parabola is f(x) = x^2 - 6x + 5, whose outputs for x = 0 through 6 are 5, 0, -3, -4, -3, 0, and 5. A second parabola is f(x) = -x^2 + 4x + 1. A launched object's height in feet t seconds after launch is h(t) = -16t^2 + 64t + 80.
PROBLEM P1: The week's parabola f(x) = x² − 6x + 5 has a = 1 and b = −6. Using the vertex formula h = −b ⁄ (2a), what is the x-value h of its vertex?
PROBLEM P2: The parabola f(x) = −x² + 4x + 1 has its vertex at x = 2. What is its maximum value k = f(2)?
PROBLEM P3: A launched object's height in feet t seconds after launch is h(t) = −16t² + 64t + 80. How many feet high is its peak?
PROBLEM P4: Complete the square to rewrite the week's parabola x² − 6x + 5 in vertex form (x − 3)² + k, using half of b, which is −3. What is k?
After the four answers, I must briefly interpret each one: what that value shows about the parabola and one use or limit of it; for P1, what h tells me about the graph, the full vertex, and whether it is a minimum or a maximum; for P2, why the value is a maximum and the range; for P3, when the peak happens as well as how high, and one limit of the model; for P4, the full vertex form, what k means and how it matches the vertex from P1, and a check of both forms at x = 0. I must then write a 3-5 sentence message for a neighbor who is new to algebra and has 40 feet of fencing for a rectangular garden with area A(w) = w(20 - w), explaining why the best width sits at the turning point, what the best width and largest area are, and one honest limit. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct parabola computations (40 points), reasoning about the vertex, its meaning, and the forms (32 points), clarity for a non-expert reader (16 points), and showing work so it can be checked (12 points). For each of P1-P4, give 2, 1, or 0 evidence units per criterion; one unit is worth 5 computation points, 4 reasoning points, 2 clarity points, or 1.5 showing-work points. Sum all four problems within each criterion. Count my final message only with P4, not again with P1-P3.
COMPUTATION UNITS: A correct vertex value, maximum, peak height, or vertex-form constant = 2; the right values and operation with only an arithmetic slip = 1; a missing result or the wrong values or operation = 0. Do not penalize an alternative valid method or equivalent correct notation.
REASONING UNITS: For P1-P3 separately, an accurate account of what that value shows about the parabola plus a use or limit tied to the week's parabola, the downward parabola, or the launch setting = 2; exactly one of the two = 1; neither = 0. For P3, a peak stated without its time or its units does not count for that part. For P4, BOTH the vertex form with the meaning of k AND a final message that explains the garden's turning point with an honest limit = 2. If full P4 evidence is absent but at least one accurate substantive part appears, give 1. No accurate reasoning = 0. An arithmetic slip alone is not a reasoning error.
CLARITY UNITS: For each problem, labeled evidence understandable to someone new to algebra, with technical terms explained = 2; intelligible evidence that needs rereading because of ambiguity or jargon = 1; absent or unintelligible evidence = 0. Include the final message with P4 if submitted. Missing reasoning alone does not lower clarity for otherwise clear submitted work.
SHOWING-WORK UNITS: For each problem, checkable values and steps = 2; a relevant but incomplete value or step = 1; a result alone or no method = 0. A visible method stays checkable despite an arithmetic slip.

Do NOT invent grading rules.

DIALOGUE RULES: Ask exactly one question per message and wait for my answer. Start by asking for my complete first attempt. Build each next question from my exact words. Probe an unsupported claim and ask me to examine its evidence. Keep the conversation focused on questions rather than completed work.

ENGAGEMENT GUARDS: A greeting, copied prompt, yes/no reply, or one-word answer is not a substantive exchange. Do not flatter or accept agreement as evidence. Praise only a concrete move I made. If evidence is missing, say what is missing and ask for it. Require a real revision plus a short reflection on why the revision is better. Do not issue the report early.

EXIT CONDITION: Continue until at least five substantive exchanges have occurred AND I have supplied all four answers and their brief interpretations AND written my message for the neighbor AND revised at least one calculation or explanation AND reflected on the revision. Stop only after whichever requirement happens last.

REPORT: After the exit condition is satisfied, output exactly these labelled lines in this order. Replace X with the exact earned score; it may end in .5 and must not be rounded to a whole number. Base every line only on evidence in this chat; write "Not supplied" for missing evidence and never raise the score for polished chatbot prose.
STUDENT'S SCORE: X/100
EVIDENCE FOR THE SCORE: [specific evidence from my calculations and reasoning]
REVISION MADE: [what I changed and why]
STRENGTH: [one earned strength]
NEXT STEP: [one concrete improvement]
Then tell me to submit this report, my revised parabola report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised parabola report with all four answers, their brief interpretations, and your message for the neighbor — written, or recorded as audio or video with a transcript, or as a labelled sketch of the parabolas with a short note; every form is graded by the same scoring criteria;
• the coach's five-line report; and
• the conversation share link.

Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text.

Equivalent route (no AI required)

This route assesses the same Week 9 skills — finding the vertex from standard form with the vertex formula, reading a vertex as a maximum or a minimum from the sign of a, finding the peak of a launched object, and converting standard form to vertex form by completing the square — for comparable time and effort, about 60 to 90 minutes, the same as the coached route. Complete the same four problems, the four brief interpretations, and the message for the neighbor. Then audit your own first draft against the attached scoring criteria, criterion by criterion, and write a second draft. Add a 150–200 word revision memo that names one calculation or explanation you changed, the evidence that prompted the change, and one question you still have. Submit the revised parabola report, your completed criteria checklist, and the revision memo in the assignment entry; together they replace the share link and the coach's report. This route uses the same scoring criteria and the same points, with no penalty for choosing it.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

The calculations, reasoning, revision, and final report must be your own. Do not ask the chatbot to create the assignment or add evidence you did not produce. Submit an honest share link from your own dialogue. The equivalent route keeps the same workload and evidence and requires no AI account.

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