Q) 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) = – x2 + 2x + 8, where x is the length (in feet) on stage level.

Based on the figure given above, answer the following questions:
(i) Determine the height of the arch.
(ii) Find zeroes of the polynomial p(x). Which points on the graph represent the zeroes?
(iii) Find the span of the arch on the stage floor.
(iv) Write the coordinates of the point of intersection of the above curve with the y-axis.

(Q 37 – 30/3/3 – CBSE 2026 Question Paper)

Ans:

Looking at the graph, the curve represents a parabola.

Since, the coefficient of x ² is negative,

Hence, the polynomial for an arch is for downwards opening parabola

(i) Height of the arch:

∵ The height of the arch corresponds to the y-coordinate of the vertex.

From the diagram, point C (1, y) is the peak of the arch,

∴ the height of the Arch will be given by value of the y-coordinate at x = 1.

∵ p(x) = – x ² + 2x + 8

∴ Value of p(x) at x = 1, p (1) = – (1) ² + 2 (1) + 8

= – 1 + 2 + 8 = 9

Therefore, Height of the arch is 9 feet.

(ii) Zeroes of the polynomial:

To find the zeroes, we need to find values of x for y = 0

∴ For p(x) = 0, – x ² + 2 x + 8 = 0

∴ x ² – 2 x – 8 = 0

By mid-term splitting, we get:

∴ x ² – 4 x + 2 x – 8 = 0

∴ x (x – 4) + 2 (x – 4) = 0

∴ (x – 4) (x + 2) = 0

∴ x = 4 and x = – 2

Therefore, the zeroes of the given polynomial are 4 and – 2.

(iii) Span of the arch:

Since, we got zeroes as 4 and – 2, it means the polynomial intersects x-axis at these values of x

∴ Intersection points shown in the diagram are: A (4,0) and B(- 2, 0)

Now, the span of the curve will be the distance between abscissa values of points A and B.

∴ Span = X coordinate of A – X coordinate of B

= 4 – ( – 2) = 4 + 2 = 6

Therefore, the span of the arch is 6 feet

(iv) Coordinates of intersection point with y-axis:

Since at y-axis, x = 0

∴ To get intersection point on y-axis, we need to calculate value of y at x = 0:

∴ p(0) = – (0) ² + 2 (0) + 8 = 8

∴ at x = 0, value of y = 8

Therefore, the intersection point with y-axis is (0, 8).

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