Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsHard
A civil engineer is designing a parabolic arch for a bridge. The height of the arch (in meters) above the ground at a horizontal distance 'x' (in meters) from the left support is given by H(x) = -0.1x^2 + 2x. What is the maximum height of the arch?
- A25 meters
- B10 meters
- C15 meters
- D20 meters
Show answer & explanationAnswer & explanation
Correct answer: B. 10 meters
First, find the x-coordinate of the vertex using x = -b / (2a). For H(x) = -0.1x^2 + 2x, a = -0.1 and b = 2. So, x = -2 / (2 * -0.1) = -2 / -0.2 = 10. Then, substitute this x-value back into the function to find the maximum height: H(10) = -0.1(10)^2 + 2(10) = -0.1(100) + 20 = -10 + 20 = 10 meters.
Why the other options are wrong
- A. This could be a result of not substituting the x-value back into the equation or a calculation error.
- C. Possible error in calculation or misinterpretation of the formula.
- D. This is the value of 'b' in the equation, not the maximum height.
Maximum/Minimum of Quadratic Function
The highest or lowest value that a quadratic function attains, occurring at the vertex of its parabolic graph.
- If the leading coefficient (a) is negative, the parabola opens downward, and the vertex is a maximum.
- If the leading coefficient (a) is positive, the parabola opens upward, and the vertex is a minimum.
- The y-coordinate of the vertex is the maximum or minimum value.
Memory trick: Find x with -b/2a, then plug it in to get the height!