Find the Ascending Region of y=-(x-2)²+1: Quadratic Function Analysis

To determine the increasing domain of the function $y = -(x-2)^2 + 1$, we'll analyze the vertex and the general behavior of parabolas.

Step-by-step solution:

• Step 1: Identify the structure of the function.

The given function is $y = -(x-2)^2 + 1$, which is a quadratic function, specifically a parabola. The general form of a parabola is $y = a(x-h)^2 + k$. Comparing, we see $a = -1$, $h = 2$, and $k = 1$. The vertex form provides key information about the parabola's orientation, position, and vertex.

• Step 2: Analyze the orientation of the parabola.

Since $a = -1$ (which is negative), the parabola opens downwards. A downward-opening parabola indicates that it decreases on either side of the vertex and increases moving towards the vertex from either direction on the x-axis.

• Step 3: Determine the vertex's location.

The vertex of the parabola is at $(h, k) = (2, 1)$. This is the maximum point for this downward-opening parabola since it opens downwards.

• Step 4: Find the increasing interval based on the vertex.

For downward-opening parabolas, the interval where the function is increasing is to the left of the vertex. Therefore, the function will be increasing for values less than the x-coordinate of the vertex.

• Step 5: Conclusion.

The interval in which the function $y = -(x-2)^2 + 1$ is increasing is $x < 2$.

Thus, the ascending area (or increasing interval) of the function is $x < 2$.