Analyze the Domains of y = -1/2x² + 1/3x - 1/4

To find the positive and negative domains of the function $y = -\frac{1}{2}x^2 + \frac{1}{3}x - \frac{1}{4}$, we must determine where the function is above or below the x-axis.

Step 1: Find the roots of the quadratic equation. This requires solving:

$-\frac{1}{2}x^2 + \frac{1}{3}x - \frac{1}{4} = 0$

Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, with $a = -\frac{1}{2}$, $b = \frac{1}{3}$, and $c = -\frac{1}{4}$, we calculate:

• Discriminant: $b^2 - 4ac = \left(\frac{1}{3}\right)^2 - 4\left(-\frac{1}{2}\right)\left(-\frac{1}{4}\right) = \frac{1}{9} - \frac{1}{2} = \frac{1}{9} - \frac{4}{8} = \frac{1}{9} - \frac{9}{18} = \frac{1 - 9}{18} = -\frac{8}{18} = -\frac{4}{9}$

The discriminant is negative, indicating no real roots.

Step 2: Analyze the parabola's orientation. Because the leading term is negative, the parabola opens downwards. With no x-intercepts, this implies the entire graph is below the x-axis.

Therefore, the function is negative for all x-values. In the context of positive and negative domains:

$x > 0 :$ none, as the function doesn't cross the x-axis in positive domain.

$x < 0 :$ all $x$, as the function is always negative.