Digital SATMath: Advanced MathEasy
A financial analyst is modeling the value of a stock, V(t), in dollars, after t months, using the function V(t) = 2t^2 - 12t + 50. When will the stock value be at its minimum?
- A32 dollars
- B3 months
- C6 months
- D50 dollars
Show answer & explanationAnswer & explanation
Correct answer: B. 3 months
The function V(t) = 2t^2 - 12t + 50 is a quadratic function, representing a parabola. Since the coefficient of t^2 (a=2) is positive, the parabola opens upwards, and its vertex represents the minimum value. The t-coordinate of the vertex is given by the formula t = -b/(2a). Here, a=2 and b=-12, so t = -(-12) / (2*2) = 12 / 4 = 3 months.
Why the other options are wrong
- A. This is the minimum stock value itself (V(3) = 2(3)^2 - 12(3) + 50 = 18 - 36 + 50 = 32), not the time.
- C. This is a distracter, possibly from incorrectly calculating 12/2 or other arithmetic errors.
- D. This is the initial stock value at t=0, not the minimum.
Vertex of a Parabola (Time)
For a quadratic function f(x) = ax^2 + bx + c, the x-coordinate of the vertex, given by -b/(2a), represents the time or input value at which the function reaches its maximum or minimum output.
- If 'a' is positive, the vertex is a minimum point.
- If 'a' is negative, the vertex is a maximum point.
- The formula -b/(2a) finds the input (e.g., time) for the extremum.
Memory trick: Time to find the 'tip-top' or 'bottom-out' of the curve!