Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsMedium
The cost C (in dollars) of producing x units of a product is given by the function C(x) = 0.5x^2 - 20x + 500. What is the minimum cost to produce units of this product?
- A$250
- B$400
- C$200
- D$300
Show answer & explanationAnswer & explanation
Correct answer: D. $300
This is a quadratic function, and its graph is a parabola. The minimum cost will occur at the vertex of the parabola. For a quadratic function ax^2 + bx + c, the x-coordinate of the vertex is given by -b/(2a). Once x is found, substitute it back into the function to find the minimum cost C(x).
Why the other options are wrong
- A. Incorrect. This could be a miscalculation of the vertex or an arithmetic error.
- B. Incorrect. This value is too high, indicating a potential error in finding the vertex or calculating the function value.
- C. Incorrect. This might be a result of a calculation error after finding the vertex.
Quadratic Function Vertex
The vertex of a parabola, which is the graph of a quadratic function, represents the maximum or minimum point of the function.
- For f(x) = ax^2 + bx + c, x-coordinate of vertex is -b/(2a).
- If a > 0, the parabola opens upward, and the vertex is a minimum.
- If a < 0, the parabola opens downward, and the vertex is a maximum.
Memory trick: Vertex formula finds the peak or valley of the curve.