Week 9 Model/Build Studio: Where Does the Fountain Jet Peak?

Part 1 — The Big Picture

This week you learned that every parabola has a turning point, the vertex, and that vertex form a(x − h)² + k puts that point in plain sight while standard form ax² + bx + c shows the y-intercept and the numbers the discriminant needs. A park fountain's water jet rises from the nozzle, tops out, and falls back to the ground in exactly that shape. In this Studio you take seven measured heights of one jet, find its vertex from the symmetry of the data, build its model in vertex form and in standard form, check the model against every measurement by substitution, see it in Desmos or GeoGebra, and decide what happens when the nozzle is raised. Then you check a chatbot's account of the same jet and catch the mistakes it makes.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: From seven measured heights of a fountain's water jet, where is the jet's highest point, what model a(x − h)² + k and ax² + bx + c fits every measurement, and what changes when the nozzle is raised 2 feet?

Workshop data: Park fountain jet heights (seven distances)

Source: A workshop-only set of measurements for one park fountain's water jet, printed in full in the second scaffold table: the distance from the nozzle along the ground and the height of the water above the ground, both in feet.

• Rows: 7 measurements; the first scaffold table has 4 rows of model-building steps and the second lists all 7 measurements
• Columns: distance from the nozzle in feet (x) and height of the water in feet (y)
• Distances: 0, 1, 2, 3, 4, 5, and 6 feet
• Heights: 0, 5, 8, 9, 8, 5, and 0 feet

Part 3 — Materials & Procedure

Tools and materials

• Desmos (free) or GeoGebra
• A notebook or document for your written work

Ways of working

• build-a-model-from-real-numbers
• apply-the-exponent-rules-step-by-step
• verify-by-substitution

Procedure

1. Open the free Desmos graphing calculator at desmos.com/calculator, or GeoGebra's graphing calculator. Click the plus sign at the top left and choose table. Type the seven distances 0, 1, 2, 3, 4, 5, 6 in the x column and the seven heights 0, 5, 8, 9, 8, 5, 0 in the y column. Tool: Desmos (free) or GeoGebra. Check: Seven dots appear that rise, level off at the top, and fall back to the ground in a symmetric arch.
2. In your notebook, find the two distances whose heights match on either side of the top and use them to locate the axis of symmetry and the vertex (h, k). Write this in the first scaffold table. Tool: A notebook or document for your written work. Check: Matching heights pair up on either side of one middle distance, which is the axis of symmetry.
3. Find a by substituting the measurement at the nozzle, x = 0 and y = 0, into y = a(x − h)² + k and solving for a one step per line. Then expand your vertex form into standard form y = ax² + bx + c, squaring first and distributing second, and check the vertex with h = −b ⁄ (2a). Tool: A notebook or document for your written work. Check: A negative value of a, a vertex form and a standard form that describe the same jet, and a vertex formula result that matches your vertex.
4. Back in the graphing calculator, type y = a(x - h)^2 + k on a new line and accept the sliders offered for a, h, and k. Drag the sliders until the curve passes through all seven dots. Then type your own standard form on another new line and click its highest point to read the vertex. Tool: Desmos (free) or GeoGebra. Check: Your typed model passes through every dot, the slider values match your a, h, and k, and the highest point the calculator labels matches your vertex.
5. Verify by substitution: for each of the seven distances, put x into your standard form, compute the height step by step with the square taken before the minus sign, and compare it with the measured height. Record each check in the second scaffold table. Tool: A notebook or document for your written work. Check: Every substituted height equals the measured height.
6. Suppose the nozzle is raised 2 feet, so every height grows by 2 feet. Write the new model in vertex form and in standard form, type it into the graphing calculator, and click where it crosses the y-axis and where it meets the x-axis. Tool: Desmos (free) or GeoGebra. Check: The new arch has the same shape and the same axis but sits 2 feet higher, crosses the y-axis above the origin, and meets the x-axis once on each side of the nozzle.

Fallback route

When to use it: Use this route if no graphing calculator or internet connection is available.

Plot the seven measurements by hand on graph paper with distance across the bottom and height up the side, sketch the smooth arch through them, and mark the highest point and the mirror line. Complete both scaffold tables, every analysis question, and the critique step from your hand graph and your notebook; submit the same items for equal credit.

This route earns the same credit as the primary route.

Part 4 — The Scaffold

Build the model one step at a time, so you can see where each number in a(x − h)² + k comes from in the measurements.

Student scaffold table 1

Step	Your work	Result
Axis of symmetry from matching heights
Vertex (h, k) and whether it is a maximum or a minimum
The value of a from the nozzle measurement
Standard form, checked with h = −b ⁄ (2a)

Check your model against every measurement, so you know it fits all seven heights and not only the ones you used to build it.

Student scaffold table 2

Distance x (feet)	Measured height (feet)	Your model's height, worked step by step	Match? (yes or no)
0	0
1	5
2	8
3	9
4	8
5	5
6	0

Part 5 — Analysis Questions

1. Show how you found the vertex from matching heights, the steps that find a from the nozzle measurement, and the expansion into standard form. Then confirm the vertex with h = −b ⁄ (2a) and k = f(h).
2. In plain words, what do the two coordinates of the vertex mean for someone standing by the fountain, with units? Explain why the vertex is a maximum here, using the sign of a.
3. Describe your substitution checks: did all seven measurements match your model? Say what it would have meant if one height had not matched.
4. Write the raised jet's model in vertex form and in standard form. Give its vertex and its y-intercept, compute its discriminant, and say how many x-intercepts it has and which of them make sense for water landing on the ground.
5. Choose one and answer it in 3–4 sentences. (a) Real water jets are pushed by wind and lose speed to the air. Explain where your parabola could stop telling the truth and what you would measure to check it. (b) A real measuring tape is only read to the nearest few inches. Explain how a small reading error in one height would show up in your substitution table and whether it would mean your model is wrong.

Part 6 — AI-Critique Moment

Choose one equal-credit critique route. Route A: ask an approved AI tool to critique your completed work, then check its response against your own evidence. Route B is the no-AI equivalent printed below.

Approved tools: Claude, Gemini, ChatGPT, Grok, Microsoft Copilot.

Prompt to paste:

I've pasted my completed Week 9 Model/Build Studio tables above. They come from a park fountain's water jet, measured at seven distances from the nozzle along the ground. The heights of the water above the ground are: 0 feet: 0 feet high; 1 foot: 5 feet; 2 feet: 8 feet; 3 feet: 9 feet; 4 feet: 8 feet; 5 feet: 5 feet; 6 feet: 0 feet. Please do these things. First, find the vertex of the jet's path, say whether it is a maximum or a minimum, and write the path in vertex form y = a(x − h)² + k and in standard form y = ax² + bx + c. Second, check your model by substituting x = 3. Third, if the nozzle is raised 2 feet so every height grows by 2 feet, write the new model, give its vertex and its y-intercept, and use the discriminant to say how many x-intercepts it has. Then answer six short review questions from earlier weeks: evaluate −2⁴ and (−2)⁴; simplify 2³ · 2⁴ and (2²)³; say whether 2⁻² is a negative number; say whether √2 and √16 are rational or irrational; expand 2(x − 1)² − 3 into standard form; and give the slope and the y-intercept of y = −2x + 5. Last, answer three preview questions from later in the course: simplify (x² − 9) ⁄ (x + 3); solve √(x + 3) = x + 1; and say whether log(x + 4) can be rewritten as log(x) + log(4).

Test the response for these likely failures:

• Chatbots often square the minus sign along with the input when they substitute into −x² + 6x, computing −3² as (−3)² = 9 and reporting a height of 9 + 18 = 27 feet at x = 3, or they evaluate −2⁴ as 16.
• Chatbots sometimes move a coefficient inside the square when expanding vertex form, writing 2(x − 1)² − 3 as (2x − 2)² − 3 = 4x² − 8x + 1, or they square the −1 of the jet's model so that −(x − 3)² becomes (x − 3)².
• Chatbots sometimes swap the two numbers of a line, calling the 5 in y = −2x + 5 the slope.
• Chatbots sometimes multiply exponents when multiplying powers, writing 2³ · 2⁴ = 2¹², or add them for a power of a power, writing (2²)³ = 2⁵.
• Chatbots sometimes say 2⁻² is a negative number, such as −4.
• Chatbots sometimes call √2 rational because a calculator shows 1.4142, or call √16 irrational because it has a root sign.
• When a chatbot rearranges an inequality on the way to an answer, it sometimes divides by a negative number without reversing the sign, for example turning −2x + 1 < 9 into x < −4.
• Chatbots sometimes cancel terms instead of factors, for example crossing out the x in (x + 3) ⁄ x to get 3.
• Chatbots sometimes keep both candidates of √(x + 3) = x + 1 without checking them in the original equation, and they sometimes keep a negative x-intercept for the raised jet as a real landing spot.
• Chatbots sometimes split the log of a sum, writing log(x + 4) = log(x) + log(4).

Write 3–5 sentences: name what the chatbot got right about the jet's model, then quote at least two of its errors (or two from the printed flawed analysis on the equivalent route), name what is wrong with each, and correct each one with evidence from your own tables, your substitution checks, or the course rule it breaks.

Equivalent route (no AI required)

No AI account is required for this Studio. Instead of the chatbot, critique this flawed analysis, printed here in full: "The jet's highest point is at x = 6, where the height is 0, so the vertex is (6, 0) and it is a minimum. At x = 3 the model −x² + 6x gives 9 + 18 = 27 feet. In vertex form the path is (x − 3)² + 9, and since a = 1 it opens up. Raising the nozzle 2 feet gives −(x − 3)² + 9, the same vertex, and the discriminant of −x² + 6x + 2 is 36 − 8 = 28, so both x-intercepts are real landing spots. Review: −2⁴ = 16 and (−2)⁴ = 16; 2³ · 2⁴ = 2¹²; 2⁻² = −4; √2 is rational because it is 1.4142; 2(x − 1)² − 3 = 4x² − 8x + 1; the slope of y = −2x + 5 is 5; and dividing −2x < 8 by −2 gives x < −4. Preview: (x + 3) ⁄ x simplifies to 3; √(x + 3) = x + 1 has the two solutions 1 and −2; and log(x + 4) = log(x) + log(4)." Mark it up: quote each flawed sentence, name what is wrong (a vertex read from the landing spot, a minimum called for a jet that opens down, a negated power squared along with its minus sign, the coefficient of vertex form dropped or squared, a raised model that never changed, a discriminant computed with a sign slip, a negative x-intercept kept as a landing spot, a product rule and power rule mix-up, a negative exponent read as a negative number, an irrational number called rational, the slope and the y-intercept swapped, an inequality sign not reversed, a term cancelled instead of a factor, an extraneous solution kept, or the log of a sum split), cite the evidence from your tables or the course rule, and write a corrected sentence beneath it; for the three preview sentences, say what the course will check later. This route assesses the same objective (Objective 6 — solve quadratic equations by factoring, the square root property, completing the square, and the quadratic formula; use the discriminant to predict the number of real solutions; and analyze quadratic functions and their graphs, including the vertex, intercepts, and maximum or minimum value) as the chatbot critique, takes comparable effort (about 30 minutes and 200–300 words), is scored by the same scoring criteria for the same credit, and its own submission artifact is the marked-up flawed analysis, posted in place of the chatbot share link and report.

Part 7 — What to Submit

• Both completed scaffold tables
• A screenshot of your graph with both jets, or a labelled hand-drawn graph
• Written answers to all five analysis questions
• The chatbot critique report with its share link, or the equivalent route's marked-up flawed analysis

Deliverable:

A short Model/Build Studio report on a fountain jet's measurements: two completed tables (the model-building steps, and seven substitution checks), the jet's model in vertex form and in standard form with its vertex named as a maximum, a graph of the jet and the raised jet or a labelled sketch, written answers to five analysis questions, and a critique of a chatbot's account of the same jet, or, on the equivalent route, a marked-up correction of the printed flawed analysis. 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. Choose a written, recorded, or visual form for your answers; every form is graded by the same scoring criteria.

Rubric

The rubric below shows how each submitted part is evaluated; bands are full, partial, or none.

Workshop evaluation criteria

Criterion	Full	Partial	None
Building the model from the measurements	The vertex is found from matching heights, a is solved from the nozzle measurement step by step, the vertex form is expanded into standard form by squaring first, and the vertex formula confirms the vertex.	A model is written, but the vertex is read without the symmetry or the formula, a is found without its steps, or the expansion has one slip.	No model is built, or the numbers are not drawn from the measurements.
Verifying by substitution	All seven distances are substituted into the model step by step with the square taken before the minus sign, every result is compared with the measured height, and the match is stated for each row.	Three to six rows are checked, or the checks are given as results without the steps.	No substitution check is shown.
Reading the vertex, the maximum, and the raised jet	Analysis questions 2, 4, and 5 are answered accurately: both vertex coordinates are read with units, the vertex is called a maximum because a is negative, the raised jet's vertex, y-intercept, and discriminant are correct with the behind-the-nozzle intercept ruled out, and a real limit of the model is explained.	The questions are attempted, but the vertex is called a minimum, a coordinate or unit is missing, the discriminant judgment lacks a reason, or the limit is missing.	The analysis questions are missing or do not address the fountain measurements.
Graph and completed tables	Both scaffold tables are complete, and a graph or labelled hand drawing shows the seven measurements on the jet's arch and the raised jet beside it, with the axes and the vertex labelled.	The tables or the graph are partly complete, or the axes or the vertex are unlabelled.	No tables and no graph are submitted.
Critique of the chatbot or of the printed flawed analysis	On either route, at least two errors are quoted and corrected with evidence from the student's own tables, substitution checks, or the course rule, and one correct point is acknowledged; both routes earn this band equally.	On either route, errors are named but corrected without evidence, or only one error is caught.	No critique appears, or pasted output is submitted without any checking.

Quality Gate

• Every vertex, coefficient, and height in the tables comes from your own calculations on the printed measurements, never from memory or a chatbot.
• The vertex is found from matching heights and confirmed with h = −b ⁄ (2a) and k = f(h).
• Every one of the seven substitution checks is written out step by step, with the square taken before the minus sign.
• The vertex is named as a maximum because a is negative, and both of its coordinates are stated with units.
• The raised jet's model, vertex, y-intercept, and discriminant rest on your own work, and an x-intercept the setting rules out is named as such.
• No number, rule, or source is invented anywhere in the submission.
• The critique quotes the chatbot or the flawed analysis and corrects at least two errors with evidence or the course rule.