Find Decreasing Intervals for y = (x+6)(x-8): Quadratic Function Analysis

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

• Step 1: Expand the function and find the standard quadratic form.
• Step 2: Differentiate the quadratic function to find critical points.
• Step 3: Determine intervals of increase or decrease by analyzing the derivative's sign.

Now, let's work through each step:
Step 1: The function is given in factored form: $y = (x + 6)(x - 8)$.
Expanding the expression gives $y = x^2 - 8x + 6x - 48$, which simplifies to $y = x^2 - 2x - 48$.

Step 2: Calculate the derivative of the expanded function: $y' = \frac{d}{dx}(x^2 - 2x - 48) = 2x - 2$.

Step 3: Set the derivative equal to zero to find the critical points: $2x - 2 = 0$.

Solving this equation, we find: $2x = 2$ which gives $x = 1$ as the critical point.

Step 4: Analyze the intervals determined by the critical point on the number line: - For $x < 1$, choose a point like $x = 0$: $y'(0) = 2(0) - 2 = -2$, which is negative, indicating the function is decreasing.

- For $x > 1$, choose a point like $x = 2$: $y'(2) = 2(2) - 2 = 2$, which is positive, indicating the function is increasing.

Therefore, the function is decreasing in the interval $x < 1$.

This analysis matches the provided correct answer, so the solution to the problem is $x < 1$.