Find Decreasing Intervals: Analyzing y = (x-9)(5-x)

To find the intervals where the function is decreasing, we'll do the following:

• Step 1: Rewrite the function in standard form.
• Step 2: Determine the direction in which the parabola opens.
• Step 3: Identify the vertex and use it to determine decreasing intervals.

Let's begin by expanding the given function:

Step 1: The function $y = (x-9)(5-x)$ can be expanded to:

$y = -(x - 9)(x - 5) = -(x^2 - 14x + 45)$

This simplifies to:

$y = -x^2 + 14x - 45$

Step 2: Analyze the parabola:

The quadratic equation $-x^2 + 14x - 45$ has a negative leading coefficient (-1), indicating that the parabola opens downward.

Step 3: Find the vertex:

The vertex of a parabola $ax^2 + bx + c$ is given by $x = -\frac{b}{2a}$. Here, $a = -1$ and $b = 14$.

Thus, the x-coordinate of the vertex is $x = -\frac{14}{2(-1)} = 7$.

Because the parabola opens downward, the function decreases after the vertex. Consequently, the function is decreasing for:

$x > 7$