https://file.aiquickdraw.com/mathbot/file/1726192714159un3olqge.jpg

The equation provided in the image is \( y = \sqrt{8x - x^2} \), which describes a semi-circle. The figure shows a graph with a clear semi-circular shape. 

### Step-by-step Explanation:
1. The given equation \( y = \sqrt{8x - x^2} \) can be rewritten to recognize the semi-circle form by completing the square. We aim to rewrite it in a form that matches the general equation of a circle:
   \[
   x^2 - 8x + r^2 = r^2
   \]
   Let's complete the square for the expression inside the square root.

2. Rewrite the expression:
   \[
   8x - x^2 = -(x^2 - 8x)
   \]
   Completing the square on \(x^2 - 8x\):
   \[
   x^2 - 8x = (x - 4)^2 - 16
   \]
   Thus, the equation becomes:
   \[
   y = \sqrt{16 - (x - 4)^2}
   \]
   
3. This represents the top half of a circle centered at \( (4, 0) \) with radius 4, i.e. the equation of a circle is:
   \[
   (x - 4)^2 + y^2 = 16
   \]
   
4. The graph shown matches this interpretation, with the semi-circle extending from \(x = 0\) to \(x = 8\), and the maximum point at \( (4, 4) \).

#### Use transformations to construct a function whose graph is as shown:
From the graph, it appears the new function involves shifting this semi-circle horizontally or vertically. 

If the graph in the figure is shifted horizontally or vertically, the equation of the new function could look something like:

\[
y = \sqrt{8(x - h) - (x - h)^2} + k
\]
Where \( h \) and \( k \) represent horizontal and vertical shifts respectively.

### Would you like more details on any part of this solution? Here are some relative questions to further explore this topic:

1. How do you complete the square for quadratic expressions?
2. What is the general form of a circle's equation and how can it be transformed?
3. How do horizontal and vertical shifts affect the graph of a function?
4. How would you graph this function by hand without plotting points?
5. What does the domain of this function look like?

#### Tip: Always check the center and radius of a semi-circle or circle from the equation by completing the square. This will help you interpret the graph easily!

Use transformations to construct a function whose graph is as shown in the figure: y = √(8x - x^2).

Learn how to transform and graph the semi-circle function y = √(8x - x^2) using key mathematical techniques such as completing the square and applying graph transformations. Suitable for high school students in Grades 9-12.

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