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 \(p(x) = -x^2 + 2x + 8\), where x is the length (in feet) on stage level.
Observe the figure and 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 maximum value of any quadratic expression \(ax^2 + bx + c\) (where \(a \lt 0\)) is also given directly by the formula:
\[ y_{\text{max}} = \frac{4ac - b^2}{4a} \]
Substituting the coefficients here gives 4(-1)(8) - 2^24(-1) = -32 - 4-4 = 9 feet. This is a direct vertex formula shortcut!

Answer 1

Step 1: Find the vertex by completing the square instead of using the $-\dfrac{b}{2a}$ formula.
\[ p(x) = -x^{2}+2x+8 = -(x^{2}-2x)+8 \]
Complete the square inside the brackets:
\[ x^{2}-2x = (x-1)^{2}-1 \]
So:
\[ p(x) = -\big[(x-1)^{2}-1\big]+8 = -(x-1)^{2}+9 \]
Since $-(x-1)^{2}$ is always zero or negative, $p(x)$ reaches its largest value when $(x-1)^{2}=0$, that is at $x=1$, giving a maximum value of $9$.

Step 2: State the height of the arch.
The maximum height of the arch is $9$ feet.

Step 3: Find the zeroes using the quadratic formula instead of factoring by inspection.
Set $p(x)=0$:
\[ -x^{2}+2x+8=0 \implies x^{2}-2x-8=0 \]
Using the quadratic formula with $a=1$, $b=-2$, $c=-8$:
\[ x = \frac{-b\pm\sqrt{b^{2}-4ac}}{2a} = \frac{2\pm\sqrt{4+32}}{2} = \frac{2\pm 6}{2} \]
\[ x = 4 \quad \text{or} \quad x = -2 \]
These correspond to the points $(4,0)$ and $(-2,0)$ where the curve touches the x-axis (the stage floor).

Step 4: Alternative part (ii)(b), the span of the arch.
The span is the distance between the two zeroes on the stage floor:
\[ \text{Span} = 4-(-2) = 6\text{ feet} \]

Step 5: Find the y-intercept.
\[ p(0) = -(0)^{2}+2(0)+8 = 8 \]
So the curve meets the y-axis at $(0,8)$.

Final Answer:
(i) The height of the arch is $9$ feet. (ii)(a) The zeroes are $4$ and $-2$, at points $(4,0)$ and $(-2,0)$. (ii)(b) The span is $6$ feet. (iii) The y-axis intersection point is $(0,8)$.
\[ \boxed{9\text{ ft},\ (4,0)\ \&\ (-2,0),\ 6\text{ ft},\ (0,8)} \]