SAT Math: Find the Maximum Value of f(x) = -2(x - 5)² + 18 in Under 5 Seconds
In vertex form f(x) = a(x - h)^2 + k, vertex is (5, 18). Since a = -2 < 0, the maximum value is 18.
Exact Question Text
What is the maximum value of the quadratic function f(x) = -2(x - 5)^2 + 18?
1. Step-by-Step Formal Solution
- Vertex is (h, k) = (5, 18).
- (x - 5)^2 >= 0 for all real x, so -2(x - 5)^2 <= 0.
- Maximum occurs at x = 5 where f(5) = 18.
2. Fast Desmos / Calculator Shortcut
Graph f(x) = -2(x - 5)^2 + 18 and click the top grey vertex point: (5, 18). Maximum value is y = 18.
3. Distractor Trap Analysis
- Trap A (Sign Flip): Forgetting to reverse inequality or dropping negative sign.
- Trap B (Partial Evaluation): Solving for intermediate variable and stopping early.
- Trap C (Extreme Assumption): Over-inferring beyond textual evidence.
- Option Correct: Accurately satisfies all algebraic and logical constraints.
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