Reflections Across Axes Determining The Axis Of Reflection For Point (-5, -9) To (5, -9)

JU07/08/2025 08, 2025 by THE IDEN 89 views

Understanding Reflections Across Axes The point (-5, -9) was reflected over an axis to become the point (5, -9). Which axis was it reflected over?

Let's delve into the concept of reflections across axes in coordinate geometry. Reflections are transformations that produce a mirror image of a point or shape over a line, which in this case is one of the coordinate axes. To accurately determine which axis a point was reflected over, we need to analyze how the coordinates change during the transformation. The essence of this question is pinpointing which coordinate (x or y) changed its sign while the other remained constant. This will directly lead us to the axis of reflection.

Decoding Coordinate Reflections

When reflecting a point across the x-axis, the x-coordinate remains the same, while the y-coordinate changes its sign. Mathematically, if a point (x, y) is reflected across the x-axis, its image becomes (x, -y). This is because the vertical distance from the point to the x-axis stays the same, but the direction is flipped. Think of it as folding the coordinate plane along the x-axis; the vertical position of the point is mirrored, but the horizontal position doesn't change. So, the point's distance from the x-axis is preserved, but its direction (above or below) is inverted. For example, if we have a point (2, 3) and reflect it across the x-axis, the new point will be (2, -3). The x-coordinate stays at 2, but the y-coordinate changes from 3 to -3. This principle is fundamental in understanding reflections over the x-axis.

Conversely, when reflecting a point across the y-axis, the y-coordinate remains unchanged, while the x-coordinate changes its sign. Thus, a point (x, y) reflected across the y-axis transforms into (-x, y). Here, the horizontal distance to the y-axis is preserved, while the direction (left or right) is reversed. Imagine folding the coordinate plane along the y-axis; the horizontal position of the point is mirrored, but the vertical position remains the same. The point maintains its vertical distance from the y-axis, but its horizontal direction is inverted. For example, consider a point (-4, 1). Reflecting it across the y-axis results in the point (4, 1). The y-coordinate remains 1, but the x-coordinate changes from -4 to 4. This understanding is crucial for grasping reflections over the y-axis.

Understanding these rules is key to solving reflection problems in coordinate geometry. By observing which coordinate changes its sign, we can easily identify the axis of reflection. Let's apply these concepts to the given question.

Analyzing the Given Points: (-5, -9) and (5, -9)

The problem presents us with two points: the original point (-5, -9) and its reflection (5, -9). Our task is to determine the axis over which the reflection occurred. To do this, we will carefully compare the coordinates of both points and identify which coordinate has changed its sign while the other remains constant.

Looking at the x-coordinates, we see that -5 has changed to 5. This indicates a change in sign for the x-coordinate. On the other hand, the y-coordinate remains constant at -9 in both points. This observation is crucial because it directly tells us which axis the reflection occurred over. Since the x-coordinate changed its sign and the y-coordinate remained the same, the reflection must have occurred over the y-axis. As we discussed earlier, reflections over the y-axis involve changing the sign of the x-coordinate while keeping the y-coordinate constant.

This is in line with the rule that a point (x, y) reflected over the y-axis becomes (-x, y). In our case, (-5, -9) becomes (5, -9), which fits this pattern perfectly. Therefore, by analyzing the change in coordinates, we can confidently conclude that the reflection was performed over the y-axis.

Evaluating the Answer Choices

Now, let's examine the provided answer choices to determine the correct one:

A. x-axis, because the x-coordinate is the opposite B. x-axis, because the y-coordinate is the opposite C. y-axis, because the x-coordinate is the opposite

Option A suggests that the reflection was over the x-axis because the x-coordinate is the opposite. This is incorrect because, for reflections over the x-axis, the y-coordinate should be the one changing sign, not the x-coordinate.

Option B also suggests the x-axis, but this time it claims the y-coordinate is the opposite. While it correctly identifies that a coordinate should be the opposite, it mistakenly associates this with the x-axis. Reflections over the x-axis would indeed change the y-coordinate's sign, but in our case, the x-coordinate is the one that changed.

Option C proposes that the reflection was over the y-axis because the x-coordinate is the opposite. This is the correct answer. As we deduced from our analysis, the change in the x-coordinate's sign while the y-coordinate remained constant is indicative of a reflection over the y-axis. Therefore, option C accurately describes the transformation that occurred.

In summary, by methodically comparing the coordinates and understanding the rules of reflections, we can confidently select the correct answer.

Conclusion

In conclusion, the point (-5, -9) was reflected over the y-axis to become the point (5, -9). This is because the x-coordinate changed its sign while the y-coordinate remained constant. Understanding the properties of coordinate reflections is crucial in solving such problems accurately. By remembering that reflections over the x-axis change the sign of the y-coordinate and reflections over the y-axis change the sign of the x-coordinate, you can easily identify the axis of reflection. This principle is a cornerstone of coordinate geometry and helps in visualizing transformations effectively.

Therefore, the correct answer is:

C. y-axis, because the x-coordinate is the opposite