GED Mathematical Reasoning TestAlgebraic Problem Solving with Graphs and FunctionsMedium
A company's quarterly profits (in thousands of dollars) are represented by the function P(q) = -2q² + 16q - 10, where q is the quarter number (q=1 for Q1, q=2 for Q2, etc.). What is the maximum profit the company achieves?
- A$24,000
- B$22,000
- C$18,000
- D$6,000
Show answer & explanationAnswer & explanation
Correct answer: B. $22,000
To find the maximum profit, we need to find the vertex of the parabola, as the function is a quadratic with a negative leading coefficient (meaning it opens downwards). The q-coordinate of the vertex is -b/(2a). Then substitute this q-value back into the function to find the maximum profit.
Why the other options are wrong
- A. This is a plausible distractor, perhaps from a calculation error.
- C. This is a plausible distractor, perhaps from a calculation error.
- D. This is a plausible distractor, perhaps from a calculation error.
Maximum/Minimum of a Quadratic Function
For a quadratic function f(x) = ax² + bx + c, the maximum or minimum value occurs at the vertex. If 'a' is negative, it's a maximum; if 'a' is positive, it's a minimum. The x-coordinate of the vertex is -b/(2a).
- Vertex x-coordinate: x = -b / (2a)
- Vertex y-coordinate (max/min value): f(-b / (2a))
- Parabola opens down (a<0) means vertex is a maximum.
- Parabola opens up (a>0) means vertex is a minimum.
Memory trick: Vertex's Point, Peak or Valley, that's the extreme!