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.
Show 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.
Step 1: Understanding the Question:
The topic of this question is Quadratic Polynomials and Parabolic Curves.
We are given a quadratic polynomial representing a parabolic arch:
\[ p(x) = -x^2 + 2x + 8 \]
We need to answer questions related to the height, zeroes, span, and y-intercept of the arch.
Step 2: Key Formula or Approach:
- The maximum height of the parabolic arch corresponds to the vertex of the parabola.
- The x-coordinate of the vertex of a parabola \( y = ax^2 + bx + c \) is given by:
\[ x = -\frac{b}{2a} \]
- The zeroes of the polynomial are the values of \( x \) for which \( p(x) = 0 \).
- The span is the distance between the two zeroes on the x-axis.
- The y-intercept is found by substituting \( x = 0 \) into \( p(x) \).
Step 3: Detailed Explanation:
1. Part (i): Determine the height of the arch:
Identify coefficients for \( p(x) = -x^2 + 2x + 8 \):
- \( a = -1, b = 2, c = 8 \)
Find the x-coordinate of the vertex:
\[ x = -\frac{b}{2a} = -\frac{2}{2(-1)} = 1 \]
The maximum height of the arch is the value of \( p(x) \) at \( x = 1 \):
\[ \text{Height} = p(1) = -(1)^2 + 2(1) + 8 = -1 + 2 + 8 = 9\text{ feet} \]
2. Part (ii)(a): Find zeroes of the polynomial and points on graph:
Set the polynomial to zero:
\[ -x^2 + 2x + 8 = 0 \implies x^2 - 2x - 8 = 0 \]
Factorize by splitting the middle term:
\[ x^2 - 4x + 2x - 8 = 0 \]
\[ x(x - 4) + 2(x - 4) = 0 \]
\[ (x - 4)(x + 2) = 0 \]
Thus, the zeroes are \( x = 4 \) and \( x = -2 \).
On the graph, the points representing the zeroes are the x-intercepts:
- Point \( A = (4, 0) \)
- Point \( B = (-2, 0) \)
3. Part (ii)(b): Find the span of the arch on the stage floor:
The span of the arch is the distance between the two x-intercept points \( A \) and \( B \):
\[ \text{Span} = x_A - x_B \]
\[ \text{Span} = 4 - (-2) = 6\text{ feet} \]
4. Part (iii): Coordinates of the intersection point with y-axis:
The intersection with the y-axis occurs when \( x = 0 \):
\[ p(0) = -(0)^2 + 2(0) + 8 = 8 \]
Therefore, the coordinates of the point of intersection are \( (0, 8) \).
Step 4: Final Answer:
(i) Height of the arch is 9 feet.
(ii)(a) Zeroes are 4 and -2, and points on graph are \(A(4, 0)\) and \(B(-2, 0)\).
(ii)(b) Span of the arch is 6 feet.
(iii) The y-intercept point is \((0, 8)\).
