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(x) = -x² + 8x - 7, where x is the quarter number (1 for Q1, 2 for Q2, etc.). During which quarter did the company achieve its maximum profit?
- AQuarter 2
- BQuarter 1
- CQuarter 3
- DQuarter 4
Show answer & explanationAnswer & explanation
Correct answer: D. Quarter 4
The function P(x) = -x² + 8x - 7 is a quadratic function, and its graph is a parabola opening downwards. The maximum profit occurs at the vertex of the parabola. The x-coordinate of the vertex is given by -b/(2a). For P(x) = -x² + 8x - 7, a = -1 and b = 8. So, x = -8 / (2 * -1) = -8 / -2 = 4. The maximum profit occurs in Quarter 4.
Why the other options are wrong
- A. P(2) = -4 + 16 - 7 = 5. Not the maximum.
- B. P(1) = -1 + 8 - 7 = 0. Not the maximum.
- C. P(3) = -9 + 24 - 7 = 8. Not the maximum.
Vertex of a Parabola
The vertex of a parabola is the highest or lowest point on its graph. For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by the formula x = -b/(2a).
- If a > 0, the parabola opens upward, and the vertex is a minimum point.
- If a < 0, the parabola opens downward, and the vertex is a maximum point.
- The y-coordinate of the vertex is found by substituting the x-coordinate back into the function.
Memory trick: Vertex is the peak or valley!