The question asks for a transformation of the function $y = \sqrt{5x - x^2}$ so that its graph matches the one shown in the image, with an inverted version of the graph, likely shifted downwards.

Let’s break this down:

1. Original Function: The given function $y = \sqrt{5x - x^2}$ is a square root function that forms a half-parabola opening upwards, as the expression inside the square root is quadratic and downward-facing.

2. Target Transformation: The graph in the image shows two key differences from the original function:

   • The top graph is similar to the original function but reflected over the x-axis.
   • The bottom graph is a reflection and a vertical translation downward.

Transformation Steps:

1. Reflection: To reflect the graph over the x-axis, you multiply the function by -1: $y = -\sqrt{5x - x^2}$ This accounts for the upside-down parabola.

2. Vertical Translation: The second part of the graph is the reflected one, but shifted downward. For the lower graph, it appears to be moved down by around 4 units. So, we modify the function by subtracting 4: $y = -\sqrt{5x - x^2} - 4$

Thus, the function after applying transformations is: $y = -\sqrt{5x - x^2} - 4$

This equation now matches the graph shown, where the top part represents the reflection, and the bottom part includes both the reflection and downward translation.

Would you like to go over any specific steps or need further clarifications?

Related Questions:

1. How can you reflect a graph over the y-axis?
2. What happens to a graph if you translate it 3 units to the right?
3. What is the effect of multiplying a function by a constant greater than 1?
4. How do you determine the domain of a square root function?
5. What transformations lead to stretching or compressing a graph?

Tip: When reflecting graphs over the axes, always remember that multiplying the output by -1 reflects over the x-axis, while altering the input reflects over the y-axis.