Find Increasing Intervals for y = -3x² + 12x - 9: Quadratic Function Analysis

To find the intervals where the function $y = -3x^2 + 12x - 9$ is increasing, we first need to determine the vertex of the parabola since it will help us identify the change in behavior from increasing to decreasing.

Step 1: Calculate the derivative of the function.
The function is $y = -3x^2 + 12x - 9$. The derivative is calculated as follows:

$y' = \frac{d}{dx} (-3x^2 + 12x - 9) = -6x + 12$

Step 2: Find the critical points by setting the derivative equal to zero:
$-6x + 12 = 0$

Solve for $x$:
$-6x = -12$
$x = 2$

This critical point $x = 2$ represents the $x$-coordinate of the vertex of the parabola.

Step 3: Determine the sign of the derivative on either side of the critical point $x = 2$.
- For $x < 2$, choose a test point, e.g., $x = 1$:
$y'(1) = -6(1) + 12 = 6 > 0$ (Positive, indicating increasing)

- For $x > 2$, choose a test point, e.g., $x = 3$:
$y'(3) = -6(3) + 12 = -6 < 0$ (Negative, indicating decreasing)

Therefore, the function is increasing on the interval where $x < 2$.

The solution to the problem is $x < 2$.