The problem in the image asks for the construction of a function using transformations, based on the graph provided.

Let's break it down:

Given Equation:

The first equation is $y = \sqrt{8x - x^2}$, which is a semicircle centered along the x-axis with a radius of 4. The domain for this function is $0 \leq x \leq 8$.

Transformation Requirement:

The goal is to construct a similar graph using a transformation.

Incorrect Attempt:

The equation provided in the second part, $y = \sqrt{8x - 2x^2}$, has been marked incorrect, likely because it does not produce the desired transformation.

Finding the Correct Transformation:

One possible transformation of the function would involve scaling or shifting. Let's attempt to modify the original function:

Transformation Idea:

A vertical stretch could involve scaling the y-values of the original graph. Since the first graph has a maximum y-value of 4, and the second graph has a maximum y-value of 8, it seems that the y-values have been scaled by a factor of 2.

Therefore, the correct transformation could be to multiply the original equation's output by 2.

This gives: $y = 2\sqrt{8x - x^2}$

This equation would stretch the y-values of the original semicircle, doubling the maximum value, thus matching the second graph.

Would you like further details or have any specific questions?

Here are 5 questions to consider next:

1. How does scaling affect the graph of a function?
2. What are the rules for applying vertical and horizontal stretches to a function?
3. How can you identify whether a transformation is vertical or horizontal?
4. Can transformations be combined, and how do they interact?
5. How can domain and range change when a function is transformed?

Tip: When transforming functions, always check the maximum and minimum values to ensure the transformation is applied correctly.