GED Mathematical Reasoning TestAlgebraic Problem Solving with Expressions and EquationsMedium
A financial analyst is modeling the quarterly profit (P) of a company, which is described by the function P(q) = -2q² + 16q - 24, where 'q' is the number of quarters since the company started. What is the maximum profit the company achieves?
- A$8
- B$32
- C$16
- D$24
Show answer & explanationAnswer & explanation
Correct answer: A. $8
The function is a downward-opening parabola, so the maximum profit occurs at the vertex. The q-coordinate of the vertex is -b/(2a) = -16/(2*-2) = 4. Substitute q=4 into the function: P(4) = -2(4)² + 16(4) - 24 = -2(16) + 64 - 24 = -32 + 64 - 24 = 8.
Why the other options are wrong
- B. This value does not result from the correct calculation of the vertex.
- C. This is the 'b' coefficient, not the maximum profit.
- D. This is the y-intercept, which is not the maximum profit.
Quadratic Function Vertex (Maximum/Minimum)
The vertex of a parabola represents the maximum or minimum value of a quadratic function. For a function in the form ax² + bx + c, the x-coordinate of the vertex is -b/(2a).
- If 'a' > 0, the parabola opens upwards, and the vertex is a minimum.
- If 'a' < 0, the parabola opens downwards, and the vertex is a maximum.
- Substitute the x-coordinate of the vertex back into the function to find the maximum/minimum value.
Memory trick: Vertex finds the peak or pit!