Finding Intervals of Increase and Decrease for y = -(x + 6.5)² - 2.25

To determine the intervals of increase and decrease for the given function $y = -\left(x + 6\frac{1}{2}\right)^2 - 2\frac{1}{4}$, we need to follow these steps:

• Step 1: Identify the vertex. The vertex form of the parabola is $y = a(x-h)^2 + k$, where $h = -6\frac{1}{2}$ and $k = -2\frac{1}{4}$. The vertex is at the point $(-6\frac{1}{2}, -2\frac{1}{4})$.
• Step 2: Determine the orientation of the parabola. Since $a = -1$, the parabola opens downwards. This implies the function decreases (or falls) from the vertex as $x$ moves away from $-6\frac{1}{2}$.
• Step 3: Define the intervals:
• To the left of the vertex (i.e., for $x < -6\frac{1}{2}$), the function increases because the parabola opens downwards.
• To the right of the vertex (i.e., for $x > -6\frac{1}{2}$), the function decreases.

Therefore, the solution for the intervals is:

$\searrow:x>-6\frac{1}{2}\\\nearrow:x<-6\frac{1}{2}$