Finding Intervals of Increase and Decrease for y = -(x + 8/9)²

To find the intervals of increase and decrease for the function $y = -\left(x+\frac{8}{9}\right)^2$, let's follow these steps:

• Step 1: Identify the vertex of the quadratic function. The vertex form is $y = -\left(x+\frac{8}{9}\right)^2$, where $h = -\frac{8}{9}$. This indicates that the vertex is at $x = -\frac{8}{9}$.
• Step 2: Determine the direction in which the parabola opens. Since the coefficient of the squared term, $a$, is negative ($a = -1$), the parabola opens downwards.
• Step 3: Analyze the behavior of the function around the vertex.

Since the parabola opens downwards:

• The function increases as $x$ approaches the vertex from the left ($x < -\frac{8}{9}$) because the curve slopes downwards towards the vertex.
• The function decreases as $x$ moves away from the vertex to the right ($x > -\frac{8}{9}$) because the curve continues to slope downwards after passing the vertex.

Therefore, the intervals of increase and decrease are:

• Increasing: $x < -\frac{8}{9}$
• Decreasing: $x > -\frac{8}{9}$

The correct answer is choice 2: $\searrow:x > -\frac{8}{9} \\ \nearrow:x < -\frac{8}{9}$