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 the function h(x) = -0.05x(x - 60). What is the maximum height of the arch?
- A60 meters
- B120 meters
- C45 meters
- D90 meters
Show answer & explanationAnswer & explanation
Correct answer: C. 45 meters
The function is a quadratic in factored form. The x-intercepts are 0 and 60. The x-coordinate of the vertex (where the maximum height occurs) is exactly halfway between the intercepts. Substitute this x-value into the function to find the maximum height.
Why the other options are wrong
- A. Incorrectly identified the maximum height as the x-intercept.
- B. Incorrectly calculated the vertex or misinterpreted the function.
- D. Made a calculation error after finding the correct x-coordinate for the vertex.
Vertex of Parabola (Factored Form)
For a quadratic function in factored form, y = a(x - p)(x - q), the x-coordinate of the vertex is the average of the x-intercepts (p and q).
- The x-intercepts are p and q.
- The x-coordinate of the vertex is (p + q) / 2.
- Substitute the x-coordinate of the vertex into the function to find the y-coordinate (maximum/minimum value).
Memory trick: Peak or valley, the vertex holds the key.