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) = 0.5t^2 - 10t + 60. At what time t, in months, does the stock reach its minimum value?
- A5 months
- B20 months
- C10 months
- D15 months
Show answer & explanationAnswer & explanation
Correct answer: C. 10 months
The function V(t) = 0.5t^2 - 10t + 60 is a quadratic function, representing a parabola opening upwards (since the coefficient of t^2 is positive). The minimum value occurs at the vertex. The t-coordinate of the vertex for a quadratic function in the form at^2 + bt + c is given by t = -b / (2a). Here, a = 0.5 and b = -10. So, t = -(-10) / (2 * 0.5) = 10 / 1 = 10.
Why the other options are wrong
- A. This might result from -b/a, or a calculation error.
- B. This might result from -b / (a/2) or another miscalculation.
- D. This is a plausible distractor, perhaps from an incorrect formula or calculation.
Vertex of a Parabola
The vertex of a parabola is the point where it changes direction, representing the maximum or minimum value of the quadratic function. For a quadratic function f(x) = ax^2 + bx + c, the x-coordinate of the vertex is given by x = -b / (2a).
- If a > 0, the parabola opens upwards and the vertex is a minimum.
- If a < 0, the parabola opens downwards and the vertex is a maximum.
- The y-coordinate of the vertex is found by substituting the x-coordinate back into the function.
Memory trick: Vertex Vibe: Find 'x' with negative b over two a!