Find the Descending Region of y=(x+5)²+5: Quadratic Function Analysis

Find the descending area of the function

$y=(x+5)^2+5$

To determine when the function $y = (x+5)^2 + 5$ is decreasing, follow these steps:

• Step 1: Identify the vertex of the parabola. In the equation $y = (x+5)^2 + 5$, rewrite as $y = (x - (-5))^2 + 5$. Thus, the vertex is at $(-5, 5)$.
• Step 2: Recognize that this form represents an upward-opening parabola because the leading coefficient of the quadratic term is positive.
• Step 3: The nature of a standard parabola with a positive leading coefficient is that it decreases as $x$ moves from left to right until reaching the vertex, and then it increases.
• Step 4: Hence, the function decreases for values of $x$ less than the vertex $x$-coordinate, which is $-5$.

Therefore, the function is decreasing for $x < -5$.

Consequently, the solution is $x < -5$.