Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsMedium
A civil engineer is designing a parabolic arch for a bridge. The height h (in meters) of the arch above the ground at a horizontal distance x (in meters) from the left support is given by h(x) = -0.01x(x - 100). What is the maximum height of the arch?
- A100 meters
- B75 meters
- C25 meters
- D50 meters
Show answer & explanationAnswer & explanation
Correct answer: C. 25 meters
The function h(x) = -0.01x(x - 100) is a quadratic function in factored form. The roots (x-intercepts) are x = 0 and x = 100. For a parabola, the x-coordinate of the vertex (where the maximum height occurs) is exactly halfway between its roots.
Why the other options are wrong
- A. This is one of the x-intercepts (where height is 0), not the maximum height.
- B. This is a plausible distractor, perhaps from a miscalculation.
- D. This is the x-coordinate of the vertex, not the maximum height.
Vertex of Parabola (Factored Form)
For a quadratic function in factored form f(x) = a(x - r1)(x - r2), the x-coordinate of the vertex is the average of the roots (r1 + r2)/2.
- Roots (x-intercepts) are r1 and r2.
- Axis of symmetry is x = (r1 + r2)/2.
- Substitute the x-coordinate of the vertex into the function to find the maximum/minimum y-value.
Memory trick: Roots Average to Vertex X, then Plug for Y.