During a theatre drama, a backdrop of building arches was used. The shape of the curve shown below can be represented by the polynomial $p(x) = -x^2 + 2x + 8$, where x is the length (in feet) on stage level.
Generated by claude-sonnet-4-6 · 2026-06-15 10:35 · grounding stimulus
Model Answer
(i) Height of the arch:
The height is the maximum value of $p(x) = -x^2 + 2x + 8$.
At $x = 1$ (vertex): $p(1) = -1 + 2 + 8 = 9$ feet.
The height of the arch is 9 feet.
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(ii) Zeroes of p(x):
$-x^2 + 2x + 8 = 0$
$x^2 - 2x - 8 = 0$
$(x-4)(x+2) = 0$
$x = 4$ or $x = -2$
Zeroes are 4 and −2.
On the graph, these are represented by points A(4, 0) and B(−2, 0) — the points where the curve crosses the X-axis.
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(iii) Intersection with y-axis:
At $x = 0$: $p(0) = 0 + 0 + 8 = 8$
The curve intersects the y-axis at (0, 8).
Source: Polynomials (Chapter 2)
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Explanation
- (i) The vertex gives the maximum (height). For $p(x)=-x^2+2x+8$, vertex is at $x = -b/2a = 1$; substitute to get 9.
- (ii) Factorise after multiplying through by −1. Zeroes = x-intercepts → points A and B on graph. Name them with coordinates.
- (iii) y-intercept = constant term of polynomial = 8, so (0, 8). Examiners want coordinates, not just the value.