Digital SATMath: Advanced MathMedium
A financial analyst is modeling the value of a stock, V(t), in dollars, after t months, using the function V(t) = -t^2 + 10t + 200. After how many months will the stock reach its maximum value?
- A5 months
- B20 months
- C10 months
- D3 months
Show answer & explanationAnswer & explanation
Correct answer: A. 5 months
The function is a downward-opening parabola, so its maximum value occurs at the vertex. For a quadratic function in the form ax^2 + bx + c, the x-coordinate of the vertex is given by -b/(2a). Here, a = -1 and b = 10, so t = -10/(2*(-1)) = -10/(-2) = 5.
Why the other options are wrong
- B. This might be the result of a calculation error or misunderstanding the vertex formula.
- C. This might be the result of incorrectly using -b/a or setting -t^2 + 10t = 0.
- D. Incorrect calculation of the vertex x-coordinate.
Vertex of a Parabola
The highest or lowest point on the graph of a quadratic function, representing the maximum or minimum value of the function.
- For f(x) = ax^2 + bx + c, the x-coordinate is -b/(2a).
- If a > 0, the parabola opens upward, and the vertex is a minimum.
- If a < 0, the parabola opens downward, and the vertex is a maximum.
Memory trick: Peak or pit, the vertex is it.