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 equation P(x) = -2x^2 + 24x - 54, where x represents the number of units sold in thousands. What is the maximum profit the company can achieve?
- A$72,000,000
- B$18,000,000
- C$54,000
- D$18,000
Show answer & explanationAnswer & explanation
Correct answer: B. $18,000,000
To find the maximum profit, we need to find the vertex of the parabola represented by the quadratic equation. The x-coordinate of the vertex gives the number of units sold for maximum profit, and substituting this value into the equation gives the maximum profit. Remember to account for the 'thousands' in the units.
Why the other options are wrong
- A. This option results from a miscalculation of the vertex or an incorrect interpretation of the scaling factor for the units sold.
- C. This option represents the y-intercept of the profit function, which is the profit when zero units are sold, not the maximum profit.
- D. This option represents the numerical value of the maximum profit before accounting for the 'thousands' in the units sold, leading to an incorrect final answer.
Quadratic Function Max/Min
For a quadratic function in the form f(x) = ax^2 + bx + c, the maximum or minimum value occurs at the vertex. If 'a' is negative, the parabola opens downwards, and the vertex is a maximum. If 'a' is positive, it opens upwards, and the vertex is a minimum.
- Vertex x-coordinate: x = -b / (2a)
- Vertex y-coordinate: Substitute x-coordinate into the function to find the maximum/minimum value.
- Coefficient 'a' determines if it's a maximum (a < 0) or minimum (a > 0).
Memory trick: Vertex Vibe: High point or Low point, always at the turn!