The image contains a mathematical function $f(x) = 3 \sqrt{-2x + 1}$, along with sections to analyze its domain, range, extrema, increasing and decreasing intervals, and end behavior. It also asks for parent function identification and transformations.

Let's go step by step:

1. Parent Function:

The given function is a transformation of the square root function $\sqrt{x}$. So, the parent function is: $f_{\text{parent}}(x) = \sqrt{x}$

2. Transformations:

The given function is $f(x) = 3 \sqrt{-2x + 1}$. We can break down the transformations applied to the parent function $\sqrt{x}$:

• Horizontal stretch/compression and reflection: The term $-2x$ indicates a horizontal reflection across the y-axis and a horizontal compression by a factor of 2.
• Translation: The $+1$ inside the square root shifts the graph horizontally.
• Vertical stretch: The coefficient 3 outside the square root indicates a vertical stretch by a factor of 3.

3. Domain:

To find the domain, we need to determine where the expression inside the square root is non-negative (since the square root of a negative number is not real): $-2x + 1 \geq 0$ Solving for $x$: $-2x \geq -1 \quad \Rightarrow \quad x \leq \frac{1}{2}$ Thus, the domain is: $(-\infty, \frac{1}{2}]$

4. Range:

Since the function has a vertical stretch and starts from a minimum point (shifted vertically), the range is all values starting from 0 (the square root of 0) and extending upward, multiplied