Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsMedium
A financial analyst is modeling the quarterly profit of a company. The profit P (in thousands of dollars) is given by the function P(x) = -2x^2 + 20x - 30, where x is the number of units sold (in hundreds). What is the maximum profit the company can achieve?
- A$50 thousand
- B$10 thousand
- C$20 thousand
- D$30 thousand
Show answer & explanationAnswer & explanation
Correct answer: C. $20 thousand
The profit function is a quadratic equation, P(x) = ax^2 + bx + c, with a negative leading coefficient (a < 0), meaning its graph is a parabola opening downwards. The maximum value occurs at the vertex of the parabola.
Why the other options are wrong
- A. This could be a miscalculation, such as finding the x-value of the vertex as 5, then multiplying by 10 (thousands) or other error.
- B. This is the y-intercept, P(0) = -30, which is a loss, not the maximum profit.
- D. This might be a miscalculation, perhaps using the wrong sign or incorrect vertex formula.
Maximum/Minimum of Quadratic Function
For a quadratic function f(x) = ax^2 + bx + c, the maximum or minimum value occurs at the vertex, whose x-coordinate is given by -b/(2a).
- If a > 0, parabola opens up, vertex is a minimum.
- If a < 0, parabola opens down, vertex is a maximum.
- Substitute x-coordinate of vertex back into function to find max/min value.
Memory trick: Vertex Vibe: Find X, then Find Y for Max/Min.