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The diagram shows the graph of the function xy=f(x) with the x-axis as an asymptote.Find the new coordinates of A on the graph xy=f(|x|) and xy=|f(x)|

To solve the problem, we need to analyze the changes in coordinates for point A \((-5, -4)\) when the graph is modified by the functions \(xy = f(|x|)\) and \(xy = |f(x)|\).

### 1. Transformation for \(xy = f(|x|)\):
For the function \(xy = f(|x|)\), we take the absolute value of \(x\), which reflects the graph about the y-axis.

- The point \(A(-5, -4)\) will be reflected across the y-axis to \(A' = (5, -4)\).
- For points on the positive side of the x-axis, \(xy = f(x)\) and \(xy = f(|x|)\) are the same. Hence, \(A'(5, -4)\) remains unchanged after the reflection.

**New coordinates of A: \( (5, -4) \)**

### 2. Transformation for \(xy = |f(x)|\):
For the function \(xy = |f(x)|\), the function value \(f(x)\) is replaced with its absolute value. This means all y-values are non-negative. 

- The y-coordinate of point A will be changed from -4 to 4.
- Thus, the new coordinate of A is \(A'' = (-5, 4)\).

**New coordinates of A: \( (-5, 4) \)**

### Summary:
- For \(xy = f(|x|)\), the new coordinates of A are \( (5, -4) \).
- For \(xy = |f(x)|\), the new coordinates of A are \( (-5, 4) \).

Would you like more details or have any questions about these transformations?

Here are some related questions to expand on this topic:
1. What would the transformation be for the point B \((5, 4)\) under \(xy = f(|x|)\)?
2. How does the graph of \(xy = |f(x)|\) generally change the behavior of the function?
3. Can you determine the coordinates of the point A under the transformation \(xy = |f(|x|)|\)?
4. What happens to the asymptotes of the function under these transformations?
5. How would you describe the impact of these transformations on the overall symmetry of the graph?

**Tip:** When dealing with absolute value transformations, think about how the graph reflects over the axes and how the sign changes impact the function values.

Explore the transformations of graphs problem involving xy = f(|x|) and xy = |f(x)|. Learn how to find new coordinates of points on the graph under these transformations with a detailed step-by-step solution. Suitable for high school students.

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