Fix the mistake: Quadratic Functions ② Maximum, Minimum, and Applications
Maximum and Minimum Values of Quadratic Functions ① Without a Restricted Domain
For y = a(x-p)² + q, when no domain is specified, the value at the vertex is the maximum or minimum value.
Example ①
Find the minimum value of y = 2(x - 1)² - 3.
Answer: Since a = 2 > 0, the graph opens upward. The minimum value is -3 at the vertex (1, -3)（when x = 1）.
Find the minimum value of y = 2(x - 1)² - 3.
Answer: Since a = 2 > 0, the graph opens upward. The minimum value is -3 at the vertex (1, -3)（when x = 1）.
Key Point
- Opens upward（a > 0）→ minimum value = q（the vertex's y-coordinate）, no maximum value
- Opens downward（a < 0）→ maximum value = q, no minimum value
Practice Problems
- Find the maximum value of y = -3(x + 2)² + 7 and the value of x at that time.
Answers and Explanations
- Answer: Maximum value 7（when x = -2）. Since the graph opens downward, the vertex gives the maximum.
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