To find the intervals of increase and decrease for the function given:

$y = x^2 - 4x$

Step 1: Find the first derivative

We take the derivative of the function $y = x^2 - 4x$ to find the slope of the function at any point.

$\frac{dy}{dx} = 2x - 4$

Step 2: Set the first derivative equal to zero

Setting the derivative equal to zero will help us find critical points, which are potential points where the function changes from increasing to decreasing or vice versa.

$2x - 4 = 0$

Solving for $x$:

$2x = 4$ $x = 2$

Step 3: Analyze the intervals

Now, we will analyze the behavior of the derivative on either side of $x = 2$. We choose test points from the intervals $(-\infty, 2)$ and $(2, \infty)$.

• For $x < 2$ (e.g., $x = 0$): $2x - 4 = 2(0) - 4 = -4$ The derivative is negative, so the function is decreasing in this interval.

• For