Find Decreasing Intervals: Analyzing y = -(x-12)² - 4 in Vertex Form

Find the intervals where the function is decreasing:

y=(x12)24 y=-(x-12)^2-4

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1

Understand the problem

Find the intervals where the function is decreasing:

y=(x12)24 y=-(x-12)^2-4

2

Step-by-step solution

The function y=(x12)24 y = -(x-12)^2 - 4 is given in vertex form y=a(xh)2+k y = a(x-h)^2 + k where a=1 a = -1 , h=12 h = 12 , and k=4 k = -4 . This tells us the vertex of the parabola is at (12,4)(12, -4). Since a=1 a = -1 is negative, the parabola opens downward.

In such a parabola, the function is increasing to the left of the vertex and decreasing to the right. The axis of symmetry is x=12 x = 12 . To the left of x=12 x = 12 , the function increases, and to the right of x=12 x = 12 , the function decreases.

Therefore, the function is decreasing when x>12 x > 12 .

Thus, the interval where the function y=(x12)24 y = -(x-12)^2 - 4 is decreasing is for x>12 x > 12 .

The correct answer to this problem is: x>12 x > 12 .

3

Final Answer

x>12 x>12

Practice Quiz

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Note that the graph of the function shown below does not intersect the x-axis

The parabola's vertex is A

Identify the interval where the function is decreasing:

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