Question

Case Study - 2 : 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. Based on the figure, answer the following questions : (i) Determine the height of the arch. (ii) (a) Find zeroes of the polynomial p(x). Which points on the graph represent the zeroes? OR (ii) (b) Find the span of the arch on the stage floor. (iii) Write the coordinates of the point of intersection of the above curve with the y-axis.

Hint:

The constant term \( c \) in a quadratic polynomial \( ax^2 + bx + c \) is always the y-intercept.
This means you can write down the y-axis intersection point \( (0, c) \) immediately without any calculation.

Answer 1

Step 1: Find the zeroes of the polynomial using the quadratic formula.
The polynomial is $p(x) = -x^2 + 2x + 8$. To find the zeroes, set $p(x) = 0$:
\[ -x^2 + 2x + 8 = 0 \] Multiply through by $-1$ so the leading coefficient is positive, which makes the formula easier to apply:
\[ x^2 - 2x - 8 = 0 \] Here $a = 1$, $b = -2$, $c = -8$. Use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] First find the discriminant:
\[ b^2 - 4ac = (-2)^2 - 4(1)(-8) = 4 + 32 = 36 \] Since $36$ is a perfect square, $\sqrt{36} = 6$. Substitute:
\[ x = \frac{2 \pm 6}{2} \] This gives two roots:
\[ x = \frac{2 + 6}{2} = 4 \quad \text{and} \quad x = \frac{2 - 6}{2} = -2 \] So the zeroes are $x = 4$ and $x = -2$. On the graph these are the points where the curve crosses the x-axis, that is $(4, 0)$ and $(-2, 0)$.

Step 2: Find the height of the arch using the fact that the vertex lies exactly midway between the two zeroes.
A parabola is a mirror-symmetric curve, and its highest (or lowest) point, the vertex, always lies exactly halfway between its two zeroes on the x-axis.
\[ x_{\text{vertex}} = \frac{4 + (-2)}{2} = \frac{2}{2} = 1 \] Substitute $x = 1$ into $p(x)$ to get the height of the arch at its peak:
\[ p(1) = -(1)^2 + 2(1) + 8 = -1 + 2 + 8 = 9 \text{ feet} \]
Step 3: Find the span of the arch.
The span is the total horizontal distance between the two points where the arch meets the stage floor, that is, between the two zeroes:
\[ \text{Span} = 4 - (-2) = 6 \text{ feet} \]
Step 4: Find where the curve meets the y-axis.
The curve meets the y-axis where $x = 0$:
\[ p(0) = -(0)^2 + 2(0) + 8 = 8 \] So the point of intersection with the y-axis is $(0, 8)$.

Final Answer:
(i) The height of the arch is 9 feet.
(ii)(a) The zeroes are 4 and $-2$, shown on the graph as the points $(4,0)$ and $(-2,0)$.
(ii)(b) The span of the arch is 6 feet.
(iii) The curve meets the y-axis at $(0, 8)$.
\[ \boxed{\text{Height} = 9 \text{ ft}, \ \text{Span} = 6 \text{ ft}, \ (0,8)} \]