Quadratic Function Analysis: Find When y = (1/2)x² - 8 is Negative

Find the positive and negative domains of the function below:

$y=\frac{1}{2}x^2-8$

Then determine for which values of $x$ the following is true:

$f(x) < 0$

Let's solve the problem by following the outlined steps:

Step 1: Solve the Quadratic Equation.
First, solve the equation $\frac{1}{2}x^2 - 8 = 0$ to find the critical values:

$\frac{1}{2}x^2 - 8 = 0$

$\frac{1}{2}x^2 = 8$

$x^2 = 16$

$x = \pm 4$

Thus, the roots are $x = 4$ and $x = -4$.

Step 2: Determine Intervals and Test Sign of Function.
The roots divide the number line into three intervals: $x < -4$, $-4 < x < 4$, and $x > 4$.

• For $x < -4$, choose a test point like $x = -5$:

$y = \frac{1}{2}(-5)^2 - 8 = \frac{1}{2} \times 25 - 8 = 12.5 - 8 = 4.5$ (Positive)

• For $-4 < x < 4$, choose a test point like $x = 0$:

$y = \frac{1}{2}(0)^2 - 8 = -8$ (Negative)

• For $x > 4$, choose a test point like $x = 5$:

$y = \frac{1}{2}(5)^2 - 8 = \frac{1}{2} \times 25 - 8 = 12.5 - 8 = 4.5$ (Positive)

Conclusion:
Therefore, the function is negative in the interval where $-4 < x < 4$.

Thus, the solution for the inequality $\frac{1}{2}x^2 - 8 < 0$ is $-4 < x < 4$.