Find Intervals of Increase for y = (x-4)(x+2): Quadratic Function Analysis

To solve this problem, we'll determine the intervals of increase by finding the derivative of the quadratic function $y=(x-4)(x+2)$.

• Step 1: Expand the function.

The function can be written in expanded form as $y = x^2 - 2x - 8$.

• Step 2: Find the derivative.

The derivative of $y = x^2 - 2x - 8$ is $y' = 2x - 2$.

• Step 3: Solve $y' = 0$ to find critical points.

Setting the derivative equal to zero gives $2x - 2 = 0$, which simplifies to $x = 1$.

• Step 4: Determine where $y' > 0$ for increasing intervals.

Analyzing the derivative $y' = 2x - 2$:

• If $x > 1$, then $y' > 0$, indicating the function is increasing.
• If $x < 1$, then $y' < 0$, indicating the function is decreasing.

Therefore, the function $y = (x-4)(x+2)$ is increasing for $x > 1$.

As a result, the interval of increase for this function is $x > 1$.