Find Intervals of Increase and Decrease for y = -x² + 1.5x - 5.25

To solve this problem, we'll begin by finding the vertex of the function, which is a parabola:

The given function is:

$y = -x^2 + 1\frac{1}{2}x - 5\frac{1}{4}$

First, identify the coefficients $a = -1$, $b = \frac{3}{2}$, and $c = -\frac{21}{4}$.

Step 1: Find the $x$-coordinate of the vertex using the formula $x = -\frac{b}{2a}$.

$x = -\frac{\frac{3}{2}}{2(-1)} = \frac{3}{4}$

Step 2: Determine the direction of the parabola.

Since $a = -1$, the parabola opens downwards.

Step 3: Use the vertex to find intervals of increase and decrease.

• The function is increasing for $x < \frac{3}{4}$ because the parabola opens downwards.
• The function is decreasing for $x > \frac{3}{4}$.

Therefore, we can conclude:

The function is increasing on the interval $(-\infty, \frac{3}{4})$ and decreasing on the interval $(\frac{3}{4}, \infty)$.

This corresponds to:

$\searrow:x>\frac{3}{4}\\\nearrow:x<\frac{3}{4}$