Question

Write the co-ordinates of point \(A\).

Answer 1

Step 1: Understanding the Concept:
Point \(A\) is the peak (vertex) of the parabolic arch. For a quadratic polynomial \(ax^{2} + bx + c\), the \(x\)-coordinate of the vertex is given by \(x = -b/(2a)\).
Step 2: Detailed Explanation:
The polynomial is \(p(x) = -0.0025x^{2} - 0.025x + 136\).
Comparing with \(ax^{2} + bx + c\):
\(a = -0.0025\), \(b = -0.025\), \(c = 136\).
Find the \(x\)-coordinate:
\[ x = \frac{-(-0.025)}{2 \times (-0.0025)} = \frac{0.025}{-0.005} = -5 \]
Now, find the \(y\)-coordinate by substituting \(x = -5\) into \(p(x)\):
\[ p(-5) = -0.0025(-5)^{2} - 0.025(-5) + 136 \]
\[ p(-5) = -0.0025(25) + 0.125 + 136 \]
\[ p(-5) = -0.0625 + 0.125 + 136 = 136.0625 \]
Step 3: Final Answer:
The co-ordinates of point \(A\) are \((-5, 136.0625)\).

Answer 2

Step 1: Understanding the Concept:
The span of the arch is the horizontal distance between the two points where the arch meets the ground (or axis), which are points \(Q\) and \(P\).
Step 2: Detailed Explanation:
From the given diagram:
Point \(Q\) is at \((-238.5, 0)\).
Point \(P\) is at \((228.5, 0)\).
The span is the distance between these two \(x\)-coordinates on the \(x\)-axis.
\[ \text{Span} = |228.5 - (-238.5)| \]
\[ \text{Span} = 228.5 + 238.5 = 467 \text{ units} \]
Step 3: Final Answer:
The span of the arch is 467 units.

Answer 3

Step 1: Understanding the Concept:
The zeroes of a polynomial are the \(x\)-values where the graph intersects the \(x\)-axis. For a quadratic polynomial, the sum of zeroes \(\alpha + \beta = -b/a\).
Step 2: Detailed Explanation:
1. Identifying zeroes from the diagram:
The graph intersects the \(x\)-axis at \(x = -238.5\) and \(x = 228.5\).
So, \(\alpha = -238.5\) and \(\beta = 228.5\).
2. Sum of zeroes from the diagram:
\[ \alpha + \beta = -238.5 + 228.5 = -10 \]
3. Verification using polynomial coefficients:
From \(p(x) = -0.0025x^{2} - 0.025x + 136\), we have \(a = -0.0025\) and \(b = -0.025\).
\[ \text{Relationship: Sum of zeroes} = \frac{-b}{a} \]
\[ \text{Sum} = \frac{-(-0.025)}{-0.0025} = \frac{0.025}{-0.0025} = -10 \]
Since both calculations yield \(-10\), the relationship is verified.
Step 3: Final Answer:
The zeroes are \(-238.5\) and \(228.5\), and the sum relationship is verified as \(-10\).

Answer 4

Step 1: Understanding the Concept:
We substitute the specific values of \(x\) into the polynomial and evaluate. A parabola is symmetric about \(x = 0\) only if the coefficient \(b = 0\).
Step 2: Detailed Explanation:
Polynomial: \(p(x) = -0.0025x^{2} - 0.025x + 136\)
1. For \(x = 100\):
\[ p(100) = -0.0025(100)^{2} - 0.025(100) + 136 \]
\[ p(100) = -0.0025(10000) - 2.5 + 136 \]
\[ p(100) = -25 - 2.5 + 136 = 108.5 \]
2. For \(x = -100\):
\[ p(-100) = -0.0025(-100)^{2} - 0.025(-100) + 136 \]
\[ p(-100) = -0.0025(10000) + 2.5 + 136 \]
\[ p(-100) = -25 + 2.5 + 136 = 113.5 \]
3. Comparison:
\(108.5 \neq 113.5\). The values are not the same.
Step 3: Final Answer:
The values are \(p(100) = 108.5\) and \(p(-100) = 113.5\). They are not the same because the parabola's axis of symmetry is \(x = -5\), not \(x = 0\).