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Let's analyze the diagram and solve the problem step by step based on the operations described:

- The original point \( A(4, 7) \) is provided in the diagram.

### a) \( R_y = x(A) = A' \)
This indicates a reflection of point \( A \) across the \( y \)-axis. To reflect a point across the \( y \)-axis, we change the sign of the \( x \)-coordinate while keeping the \( y \)-coordinate the same.

Thus:
\[
A'(x', y') = (-4, 7)
\]

### b) \( R_y = -x(A') = A'' \)
This step indicates a reflection across the \( x \)-axis, but applied to the previously reflected point \( A' \). To reflect a point across the \( x \)-axis, we change the sign of the \( y \)-coordinate while keeping the \( x \)-coordinate the same.

Thus:
\[
A''(x'', y'') = (-4, -7)
\]

### c) \( R_y = x(A'') = A''' \)
This indicates another reflection across the \( y \)-axis, but this time applied to \( A'' \). So, we again change the sign of the \( x \)-coordinate while keeping the \( y \)-coordinate the same.

Thus:
\[
A'''(x''', y''') = (4, -7)
\]

### d) \( R_y = -x(A''') = A'''' \)
Finally, we reflect the point \( A''' \) across the \( x \)-axis. We change the sign of the \( y \)-coordinate while keeping the \( x \)-coordinate the same.

Thus:
\[
A''''(x'''', y'''') = (4, 7)
\]

### e) How are \( A \) and \( A'''' \) related?
We can see that \( A'''' \) has the same coordinates as the original point \( A \). Therefore, \( A'''' \) is identical to \( A \). This means that after four transformations, the point returns to its original position, making these reflections symmetric.

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Would you like more details on any of these reflections or have any questions? Here are five additional questions to deepen your understanding:

1. What is the general rule for reflecting a point across the \( x \)-axis and the \( y \)-axis?
2. How would the coordinates change if the point was initially located in the third quadrant?
3. What is the mathematical significance of combining multiple reflections across the axes?
4. Can you think of other transformations (e.g., rotations) that could also bring a point back to its original position after multiple steps?
5. How do the operations described differ from a 180-degree rotation around the origin?

**Tip:** When reflecting points across axes, remember that reflections are symmetrical and each operation affects either the \( x \)- or \( y \)-coordinate, not both at once.

Use the diagram to the right. Find the coordinates of the given point A(4, 7) after applying a series of reflections across the x-axis and y-axis.

Discover how to find the coordinates of point A(4, 7) after multiple reflections across the x-axis and y-axis. This problem explores reflection symmetry and coordinate geometry concepts suitable for high school students in Grades 9-11.

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