Find Decreasing Intervals for the Quadratic Function: y = 3x² - 6x + 4

To find where the function $y = 3x^2 - 6x + 4$ is decreasing, we first need to find its derivative. This involves the following steps:

• Step 1: Compute the derivative of the function.

The derivative of the function, using the formula for the derivative of a quadratic function $ax^2 + bx + c$, is given by:

$y' = \frac{d}{dx}(3x^2 - 6x + 4) = 6x - 6$

• Step 2: Set the derivative less than zero to find the decreasing interval.

We set the derivative $6x - 6$ less than zero:

$6x - 6 < 0$

Simplifying the inequality, we get:

$6x < 6$

$x < 1$

• Step 3: Identify the interval for decreasing behavior.

The inequality $x < 1$ indicates the interval where the function is decreasing. Note that this function represents a parabola that opens upwards, and hence, it decreases as $x$ approaches the vertex from the left and increases as $x$ moves right after the vertex.

Therefore, the interval where the function $y = 3x^2 - 6x + 4$ is decreasing is $x < 1$.