Analyze the Quadratic Function: Determine Positive and Negative Domains of y = -1/3x² + 2x - 4

To solve this problem, we'll follow these steps:

• Step 1: Identify roots of the quadratic equation.
• Step 2: Determine the intervals created by these roots.
• Step 3: Test each interval to see where the function is positive or negative.

Now, let's work through each step:

Step 1: We need to find the roots of the equation $-\frac{1}{3}x^2 + 2x - 4 = 0$. Using the quadratic formula:
For $ax^2 + bx + c = 0$, the roots are given by:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Here, $a = -\frac{1}{3}$, $b = 2$, and $c = -4$.

Step 2: Calculate the discriminant:
$b^2 - 4ac = 2^2 - 4(-\frac{1}{3})(-4) = 4 - \frac{16}{3} = \frac{12}{3} - \frac{16}{3} = -\frac{4}{3}$.

Since the discriminant is negative, the quadratic does not have real roots. Therefore, the function does not cross the x-axis and remains entirely above or below the x-axis.

Step 3: Analyze the leading coefficient. The quadratic function opens downwards because the leading coefficient $a = -\frac{1}{3}$ is negative. Therefore, since there are no x-intercepts, the function is negative for all $x$.

Thus, we find that:
- The positive domain of $y$ is: none.
- The negative domain of $y$ is: for all $x$.

Therefore, the solution to the problem is:

$x < 0 :$ for all $x$

$x > 0 :$ none