Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsMedium
A civil engineer is designing a road over a hill. The height of the road, h(x), in meters, relative to the ground, can be modeled by the quadratic function h(x) = -0.02x^2 + 1.2x + 5, where x is the horizontal distance in meters from a reference point. What is the maximum height of the road above the ground?
- A5 meters
- B38 meters
- C30 meters
- D23 meters
Show answer & explanationAnswer & explanation
Correct answer: D. 23 meters
For a quadratic function in the form ax^2 + bx + c, the x-coordinate of the vertex (which corresponds to the maximum or minimum) is given by -b/(2a). Here, a = -0.02 and b = 1.2. So, x = -1.2 / (2 * -0.02) = -1.2 / -0.04 = 30. Now, substitute x=30 into the function to find the maximum height: h(30) = -0.02(30)^2 + 1.2(30) + 5 = -0.02(900) + 36 + 5 = -18 + 36 + 5 = 23. Thus, the maximum height is 23 meters.
Why the other options are wrong
- A. This is the y-intercept, the height at x=0, not the maximum height.
- B. This could result from a calculation error, such as signing errors or miscalculation of the square.
- C. This is the x-coordinate at which the maximum height occurs, not the maximum height itself.
Vertex of a Parabola (Standard Form)
For a quadratic function in standard form f(x) = ax^2 + bx + c, the vertex is the point (h, k) where h = -b/(2a) and k = f(h). This vertex represents the maximum point if a < 0 or the minimum point if a > 0.
- The x-coordinate of the vertex is -b/(2a).
- The y-coordinate of the vertex is f(-b/(2a)).
- If 'a' is negative, the parabola opens downward, and the vertex is a maximum.
- If 'a' is positive, the parabola opens upward, and the vertex is a minimum.
Memory trick: Vertex formula: find the peak, then the height you seek!