Question

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

Answer 1

Step 1: Understanding the Concept:
The peak of the parabolic arch is the vertex of the quadratic function.
Instead of directly using the formula −b/2a, we convert the quadratic into vertex form by completing the square.

Given polynomial:
p(x) = −0.0025x² − 0.025x + 136

Step 2: Factor Out the Coefficient of x²:
p(x) = −0.0025 (x² + 10x) + 136

Step 3: Complete the Square:
Take half of 10 and square it:
(10/2)² = 25

Add and subtract 25 inside the bracket:
p(x) = −0.0025 (x² + 10x + 25 − 25) + 136

= −0.0025 [(x + 5)² − 25] + 136

Step 4: Simplify:
= −0.0025(x + 5)² + 0.0025 × 25 + 136
= −0.0025(x + 5)² + 0.0625 + 136
= −0.0025(x + 5)² + 136.0625

Step 5: Identify the Vertex:
The vertex form is:
p(x) = a(x − h)² + k

Here,
h = −5
k = 136.0625

So the vertex (peak point A) is:
(−5, 136.0625)

Final Answer:
The coordinates of point A are (−5, 136.0625).

Answer 2

Step 1: Understanding the Concept:
The span of an arch is the horizontal distance between the two points where the arch touches the ground.
These points are the x-intercepts of the graph.

Step 2: Identifying the Intercepts:
From the diagram:
Q = (−238.5, 0)
P = (228.5, 0)

Since both points lie on the x-axis, their y-coordinates are zero.
So the span is simply the difference between their x-coordinates.

Step 3: Calculating the Distance:
Span = |228.5 − (−238.5)|
= 228.5 + 238.5
= 467 units

Alternative View:
Distance between two points on x-axis = |x₂ − x₁|
= |228.5 − (−238.5)|
= 467

Final Answer:
The span of the arch is 467 units.

Answer 3

Step 1: Understanding the Concept:
The zeroes of a polynomial are the values of x where the graph cuts the x-axis.
For a quadratic polynomial ax² + bx + c:
Sum of zeroes (α + β) = −b/a

Step 2: Identifying Zeroes from the Graph:
From the diagram, the graph intersects the x-axis at:
x = −238.5 and x = 228.5

So,
α = −238.5
β = 228.5

Step 3: Calculating Sum from the Graph:
α + β = −238.5 + 228.5
= −10

Step 4: Verifying Using Coefficients:
Given polynomial:
p(x) = −0.0025x² − 0.025x + 136

Here,
a = −0.0025
b = −0.025

Using formula:
Sum of zeroes = −b/a

= −(−0.025) / (−0.0025)
= 0.025 / (−0.0025)
= −10

Step 5: Conclusion:
The sum obtained from the graph and from the formula are equal.
Hence, the relationship between zeroes and coefficients is verified.

Final Answer:
The zeroes are −238.5 and 228.5.
Their sum is −10, which verifies the formula −b/a.

Answer 4

Step 1: Understanding the Polynomial:
Given polynomial:
p(x) = −0.0025x² − 0.025x + 136

To compare p(100) and p(−100), we substitute the values directly.
Since the polynomial contains a linear term (−0.025x), it is not symmetric about x = 0.

Step 2: Calculating p(100):
p(100) = −0.0025(100)² − 0.025(100) + 136
= −0.0025(10000) − 2.5 + 136
= −25 − 2.5 + 136
= 108.5

Step 3: Calculating p(−100):
p(−100) = −0.0025(−100)² − 0.025(−100) + 136
= −0.0025(10000) + 2.5 + 136
= −25 + 2.5 + 136
= 113.5

Step 4: Checking Symmetry (Alternative Explanation):
Axis of symmetry of a quadratic ax² + bx + c is:
x = −b / (2a)

Here,
a = −0.0025
b = −0.025

x = −(−0.025) / (2 × −0.0025)
= 0.025 / (−0.005)
= −5

So the parabola is symmetric about x = −5, not x = 0.
Hence p(100) ≠ p(−100).

Final Answer:
p(100) = 108.5
p(−100) = 113.5
The values are not equal because the axis of symmetry is x = −5.