Find Decreasing Intervals for the Quadratic Function y = x² - 8x - 20

To determine the intervals where the function $y = x^2 - 8x - 20$ is decreasing, follow these steps:

• Step 1: Find the derivative of the function.
  The derivative of $y = x^2 - 8x - 20$ is $y' = 2x - 8$.
• Step 2: Solve for critical points using the derivative.
  Set the derivative equal to zero: $2x - 8 = 0$.
  Solve for $x$: $2x = 8 \Rightarrow x = 4$.
  This is the vertex of the parabola.
• Step 3: Analyze the sign of the derivative.
  The derivative $y' = 2x - 8$ changes sign at $x = 4$.
  - For $x < 4$, $2x - 8 < 0$ (derivative is negative), meaning the function is decreasing.
  - For $x > 4$, $2x - 8 > 0$ (derivative is positive), meaning the function is increasing.
• Step 4: Determine the interval of decrease.
  The function is decreasing on the interval $x < 4$.

Therefore, the function is decreasing for $x < 4$, which corresponds to the interval $(-∞, 4)$. This matches the correct answer: $x < 4$.