Quadratic Function Analysis: Finding Domains and Negative Regions of (1/2)x² - 12½

To find the values of $x$ for which the function $y = \frac{1}{2}x^2 - 12\frac{1}{2}$ is negative, follow these steps:

• Step 1: Solve the equation
  $y = \frac{1}{2}x^2 - \frac{25}{2} = 0$. Multiply through by 2 for simplicity:
• 
  $x^2 - 25 = 0$


• Step 2: Factor the quadratic expression:
• 
  $(x - 5)(x + 5) = 0$


• Step 3: Find the roots:
• 
  $x = 5$ and $x = -5$


• Step 4: Test intervals around these roots to find where $y < 0$.
• The intervals to test are $(-\infty, -5)$, $(-5, 5)$, and $(5, \infty)$.
• Step 5: Evaluate a test point within each interval:
• For $x = 0$ in interval $(-5, 5)$:
• $y = \frac{1}{2}(0)^2 - \frac{25}{2} = -\frac{25}{2}$, which is less than 0.


• For $x = 6$ in interval $(5, \infty)$:
• $y = \frac{1}{2}(6)^2 - \frac{25}{2} = \frac{11}{2}$, which is greater than 0.


• For $x = -6$ in interval $(-\infty, -5)$:
• $y = \frac{1}{2}(-6)^2 - \frac{25}{2} = \frac{11}{2}$, which is greater than 0.

Thus, the function is negative in the interval $-5 < x < 5$.

Therefore, the solution is $-5 < x < 5$.