Find the range of the quadratic function and the maximum or minimum value of the function. Identify the intervals on which the function is increasing or decreasing. \( y=(x-6)^{2}-3 \)

The given function is   y = (x - 6)² - 3 This is in the vertex form y = a(x - h)² + k where (h, k) is the vertex. 1. Minimum or Maximum Value:  • Since the coefficient of (x - 6)² is positive (a = 1), the parabola opens upward.  • This means the function has a minimum value.  • The minimum value occurs at the vertex. Here, the vertex is (6, -3).  • Therefore, the minimum value is y = -3 at x = 6. There is no maximum value because the function increases without bound. 2. Range:  • Because the minimum value is -3 and the function extends to infinity, the range of y is:   y ∈ [-3, ∞) 3. Intervals of Increase and Decrease:  • For x < 6: As x approaches 6 from the left, the value of (x - 6)² decreases. Therefore, the function is decreasing.  • For x > 6: As x increases from 6, the value of (x - 6)² increases. Therefore, the function is increasing. In summary: • The minimum value is -3 at x = 6. • The range is [-3, ∞). • The function is decreasing on (–∞, 6] and increasing on [6, ∞).