Find Intervals of Increase and Decrease for y = (x - 8⅙)² - 1

To determine the intervals of increase and decrease for the function $y = \left(x - 8\frac{1}{6}\right)^2 - 1$, we'll follow these steps:

• Identify the vertex of the parabola.
• Determine the direction the parabola opens (upwards for $a > 0$).
• Determine intervals based on the axis of symmetry at the vertex.

Step 1: Identify the vertex. The given function is in vertex form $y = (x - h)^2 + k$, where $h = 8\frac{1}{6}$ and $k = -1$. Thus, the vertex is $\left(8\frac{1}{6}, -1\right)$.

Step 2: Determine direction. The coefficient $a = 1$ is positive, so the parabola opens upwards. This implies the function decreases before the vertex and increases after the vertex.

Step 3: Determine intervals of increase and decrease. Since the parabola reaches a minimum at the vertex $x = 8\frac{1}{6}$:
- The function is decreasing for $x < 8\frac{1}{6}$.
- The function is increasing for $x > 8\frac{1}{6}$.

Therefore, the intervals of increase and decrease are as follows:
Decreasing interval: $x < 8\frac{1}{6}$.
Increasing interval: $x > 8\frac{1}{6}$.

The correct answer is:
$\searrow:x<8\frac{1}{6}\\\nearrow:x>8\frac{1}{6}$.