# The Vanishing Vertex: A Quadratic Function Puzzle Challenge

This puzzle asks you to reconstruct a full quadratic function from partial graph information, the same analytical work required when identifying vertex, intercepts, direction of opening and axis of symmetry from a quadratic function ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17778)). Recovering the equation from its roots draws on factoring polynomial expressions ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17775)) and on solving problems that involve quadratic equations ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17779)).

![Figure 1: A sheet of graph paper resting on a wooden desk beside a mechanical pencil, a clear plastic ruler and a scientific calculator, with a faint](https://goa-cc-uat-aili-app-001.azurewebsites.net/api/generate/e962d83c-4b04-47c6-96e6-3e4081d71e46/asset/1388)

## The puzzle

A parabola crosses the x-axis at x = 1 and x = 5. It opens downward, and its y-intercept is (0, −10).

Without graphing, determine:

1. The coordinates of the vertex.
2. The equation of the parabola written in the form y = a(x − p)² + q.
3. The maximum value of the function.

## Hints

1. The axis of symmetry of a parabola always lies exactly halfway between its two x-intercepts.
2. Write the function in factored form, y = a(x − 1)(x − 5), then use the given y-intercept to solve for a.
3. The axis of symmetry is x = 3. Substitute x = 0 into y = a(x − 1)(x − 5) and set the result equal to −10 to find a. Then substitute x = 3 into the factored equation to find the vertex's y-value.

## Answer key

**Vertex:** (3, 8)

**Equation in vertex form:** y = −2(x − 3)² + 8

**Maximum value:** 8

**Reasoning:**

The x-intercepts, x = 1 and x = 5, give the factored form y = a(x − 1)(x − 5), an application of factoring polynomial expressions ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17775)). The axis of symmetry sits halfway between the roots, at x = (1 + 5)/2 = 3.

Substituting the y-intercept (0, −10) into the factored form solves for a:

−10 = a(0 − 1)(0 − 5)
−10 = a(−1)(−5)
−10 = 5a
a = −2

The negative value of a confirms the parabola opens downward, consistent with the given description and with the direction of opening identified when analyzing quadratic functions ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17778)).

The full factored equation is y = −2(x − 1)(x − 5).

To find the vertex, substitute x = 3 (the axis of symmetry) into this equation:

y = −2(3 − 1)(3 − 5)
y = −2(2)(−2)
y = 8

The vertex is (3, 8), and since a is negative, this point is a maximum. The maximum value of the function is 8, occurring at x = 3.

Writing the equation in vertex form, y = a(x − p)² + q, using vertex (3, 8) and a = −2, gives:

y = −2(x − 3)² + 8

This matches the structure required when analyzing quadratic functions of the form y = a(x − p)² + q to determine vertex, direction of opening and axis of symmetry ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777)).

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*AILI game · language en · model claude-sonnet-5 · generated 2026-09-17 · id e962d83c-4b04-47c6-96e6-3e4081d71e46*

### Sources

- node:n1: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777)
- node:n2: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17776)
- node:n3: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17778)
- node:n4: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777)
- node:n5: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17779)
- node:n6: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17775)
- node:n7: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17782)
- node:n8: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17780)
- node:n9: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17781)
- node:n10: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17776)
- node:n11: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17785)
- resource:r1: Resources › type#studentsupport, type#teachersupport › MAT2791 (https://goa-cc-uat-aili-app-001.azurewebsites.net/library/resource/L-Uj1j9H4kypxiv587R6aA)
- resource:r2: Resources › type#studentsupport, type#teachersupport › MAT2791 (https://goa-cc-uat-aili-app-001.azurewebsites.net/library/resource/9Im-lhmbb0m3jiTVElTJ2A)
- resource:r3: Resources › type#studentsupport, type#teachersupport › MAT2791 (https://goa-cc-uat-aili-app-001.azurewebsites.net/library/resource/L2B8wqn5oUawLk4_WJDe0Q)
- resource:r4: Resources › type#studentsupport › MAT2791 (https://goa-cc-uat-aili-app-001.azurewebsites.net/library/resource/byh431Gvkk2zgLQtZNOBlA)
- resource:r5: Resources › type#studentsupport, type#teachersupport › MAT2791 (https://goa-cc-uat-aili-app-001.azurewebsites.net/library/resource/Ev9xj275wEyw23tOXVmz4w)