Week 9 — Finding the Turning Point: A Short Parabola Report — Traditional alternative (no AI)

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

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 week you learned to read a parabola's vertex and axis of symmetry, to decide whether a vertex is a minimum or a maximum from the sign of a, to find the vertex from standard form with the vertex formula, to find intercepts, and to switch between vertex form and standard form. 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 turning point. In this assignment you work four short problems, interpret each one, and explain to someone who has never studied algebra why the best garden sits at a turning point. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles.

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.

Show your arithmetic. A reader should be able to reproduce every number you report from the values above without guessing which values or steps you used.

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.

Deliverables

Submit a single parabola report containing:

• the four answers, each with the arithmetic shown and a brief interpretation of what that value shows about the parabola and one use or limit of it;
• your message for the neighbor, explaining why the best width sits at the turning point of A(w) = w(20 − w), what the best width and the largest area are, and one honest limit; and
• a one-line restatement of the values you used, so your results are reproducible.

Keep it to roughly one page. Clear and correct beats long.

Choose how you show it. Present your report as written text, or as a recorded audio or video explanation of about three minutes with a transcript, or as a labelled sketch of the parabolas with a short note. Every form is graded by the same scoring criteria. 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.

How you'll be graded

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

Academic integrity and AI use

This version is designed to be completed without AI: do the computations, the interpretations, and the writing yourself, without a chatbot or other AI tool. You may use a calculator or a free graphing calculator such as Desmos to check your arithmetic and your parabolas. Submitting AI-generated work as your own, or copying another student's numbers, is an integrity violation under the course policy.

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