Finding Increasing Intervals for y = (5-x)²: Quadratic Function Analysis

To determine the intervals where the function $y = (5-x)^2$ is increasing, we follow these steps:

• Step 1: Identify the vertex and axis of symmetry
  The function is given as $y = (5-x)^2$. Rewriting in a standard form gives $y = (x-5)^2$, indicating that this is a standard parabola shifted 5 units to the right.
• Step 2: Determine the direction of the parabola
  The standard form $y = (x-5)^2$ shows a parabola opening downwards. Its vertex is at $x = 5$ and $y = 0$, which serves as the axis of symmetry.
• Step 3: Use calculus for confirmation
  Find the derivative $\frac{dy}{dx} = 2(5-x)(-1) = -2(5-x)$. Simplifying gives $\frac{dy}{dx} = 2(x-5)$. This derivative is zero at the vertex $x = 5$. The function is increasing when $\frac{dy}{dx} > 0$, which occurs when $x > 5$.

From both the vertex and the derivative analysis, the function $y = (5-x)^2$ is increasing when $x > 5$.

Therefore, the interval where the function is increasing is $x > 5$.