Question

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:

1. Determine the height of the arch.   [1]
2. 1. Find zeroes of the polynomial p(x). Which points on the graph represent the zeroes?   [2]
                                 OR
   2. Find the span of the arch on the stage floor.   [2]
3. Write the coordinates of the point of intersection of the above curve with the y-axis.   [1]

Answer 1

Given: p(x) = −x² + 2x + 8

Where x is the length (in feet) on the stage level, and p(x) is the height of the arch.

(i) Determine the height of the arch.

The height of the arch is the maximum value of the parabola.

For a quadratic ax² + bx + c, the vertex occurs at:

x = `-b/(2a)`

x = `-2/(2(-1))`

= `-2/(-2)`

= 1

p(x) = −x² + 2x + 8

p(1) = −(1)² + 2(1) + 8

= −1 + 2 + 8

= 9

∴ Height of the arch = 9 feet

(ii) (a) Zeroes are the values of x for which p(x) = 0.

−x² + 2x + 8

Multiply both sides by −1 to simplify.

x² − 2x − 8 = 0

x² − 4x + 2x − 8 = 0

x(x − 4) + 2(x − 4) = 0

(x + 2)(x − 4) = 0

x + 2 = 0 or x − 4= 0

x = −2 or x = 4

These zeroes correspond to the points where the arch meets the stage floor (on the x-axis).

From the graph, these points are A(−2, 0) and B(4, 0).

OR

(ii) (b) The span is the distance between the two zeroes on the x-axis.

 Span = distance between the zeroes

= 4 − (−2) 

= 6 feet

(iii) Point of intersection with the y-axis.

At y-axis, x = 0

p(0) = −(0)² + 2(0) + 8

= 0 + 0 + 8

= 8

So, the point of intersection is (0, 8).