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: Understanding the Question:
We are given a quadratic polynomial representing a parabolic arch:
\[ p(x) = -x^2 + 2x + 8 \] We need to analyze this parabola to find its vertex height, roots, span on the floor, and y-intercept.

Step 2: Key Formula or Approach:
1. Height: The maximum point of a downward parabola \(ax^2 + bx + c\) occurs at the vertex:
\[ x = -\frac{b}{2a} \]
2. Zeroes: Solve the quadratic equation \(p(x) = 0\).
3. Span: The horizontal distance between the two zeroes of the polynomial on the x-axis.
4. y-intercept: Substitute \(x = 0\) to find the point where the curve intersects the y-axis.

Step 3: Detailed Explanation:
1. Part (i): Height of the arch:
- Identify coefficients: \(a = -1, b = 2, c = 8\).
- The x-coordinate of the vertex is:
\[ x = -\frac{2}{2(-1)} = 1 \]
- The maximum height is \(p(1)\):
\[ p(1) = -(1)^2 + 2(1) + 8 = -1 + 2 + 8 = 9\text{ feet} \]
2. Part (ii)(a): Zeroes of the polynomial:
- Set \(p(x) = 0\):
\[ -x^2 + 2x + 8 = 0 \implies x^2 - 2x - 8 = 0 \]
- Factorize the quadratic equation:
\[ (x - 4)(x + 2) = 0 \]
\[ x = 4 \quad \text{and} \quad x = -2 \]
- The zeroes are \(4\) and \(-2\). On the graph, these are represented by points \(A(4, 0)\) and \(B(-2, 0)\) on the x-axis.
3. Part (ii)(b) (Alternative): Span of the arch:
- The span is the distance between the two root points on the x-axis: \(B(-2, 0)\) and \(A(4, 0)\).
- Distance:
\[ \text{Span} = x_{\text{right}} - x_{\text{left}} = 4 - (-2) = 6\text{ feet} \]
4. Part (iii): Y-intercept:
- Substitute \(x = 0\) into the polynomial:
\[ p(0) = -(0)^2 + 2(0) + 8 = 8 \]
- The coordinates of the point of intersection with the y-axis are \((0, 8)\).

Step 4: Final Answer:
(i) The height of the arch is \(9\text{ feet}\).
(ii)(a) The zeroes are \(4\) and \(-2\), corresponding to points \((4, 0)\) and \((-2, 0)\).
(ii)(b) The span of the arch is \(6\text{ feet}\).
(iii) The coordinates of the y-axis intersection are \((0, 8)\).