Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsHard
A civil engineer is designing a suspension bridge. The main cable's height h (in meters) above the road at a horizontal distance x (in meters) from the left tower is modeled by the function h(x) = 0.005x^2 - 0.6x + 20. What is the minimum height of the cable?
- A5 meters
- B20 meters
- C8 meters
- D2 meters
Show answer & explanationAnswer & explanation
Correct answer: A. 5 meters
The function h(x) = 0.005x^2 - 0.6x + 20 is a quadratic function, representing a parabola opening upwards (since a = 0.005 > 0). The minimum height occurs at the vertex. First, find the x-coordinate of the vertex: x = -b / (2a) = -(-0.6) / (2 * 0.005) = 0.6 / 0.01 = 60. Now, substitute x = 60 into the function to find the minimum height: h(60) = 0.005(60)^2 - 0.6(60) + 20 = 0.005(3600) - 36 + 20 = 18 - 36 + 20 = 2.
Why the other options are wrong
- B. Incorrect. This is the y-intercept (initial height at x=0), not the minimum height.
- C. Incorrect. This would be if the vertex calculation was incorrect, or if the x-value used was different.
- D. Correct. The minimum height is 2 meters, calculated by finding the vertex's y-coordinate.
Minimum/Maximum of Quadratic Function
For a quadratic function f(x) = ax^2 + bx + c, the minimum or maximum value is the y-coordinate of the vertex, found by calculating f(-b/(2a)). It's a minimum if a>0 and a maximum if a<0.
- Vertex formula: x = -b / (2a).
- Substitute x-coordinate into f(x) to find y-coordinate.
- Positive 'a' means minimum, negative 'a' means maximum.
Memory trick: Find the X for the peak/valley, then plug it back in for the Y-value height!