Identify the following composition transformation mapping: (x,y)→(−x,−y+5)

Question

(
𝑥
,
𝑦
)
→
(
−
𝑥
,
−
𝑦
+
5
)
.

Option #1: rotate 180 degrees, then down 5

Option #2: reflect across the x-axis, then up 5

Option #3: reflect across the y-axis, then up 5

Option #4: rotate 180 degrees, then up 5

Answer 1

To analyze the transformation \( (x, y) \to (-x, -y + 5) \), let's break it down into two parts:

1. Reflection across the x-axis: The transformation from \( (x, y) \) to \( (x, -y) \) reflects each point across the x-axis.

   • After this reflection, the coordinates become \( (x, -y) \).

2. Translation upwards: After reflecting across the x-axis, we increase the y-coordinate by 5. This changes \( (x, -y) \) to \( (x, -y + 5) \).

Now, let's analyze the complete transformation step-by-step:

• Start with the point \( (x, y) \).
• Reflect across the x-axis: \( (x, y) \to (x, -y) \).
• Translate up by 5: \( (x, -y) \to (x, -y + 5) \), which changes the y-coordinate while keeping the same x-coordinate.

Finally, from the starting point \( (x, y) \) to the final transformation:

1. The x-coordinate is negated: \( -x \).
2. The y-coordinate has been transformed to \( -y + 5 \).

So the complete transformation can be viewed as:

• Reflect across the x-axis
• Translate up by 5

Thus, the correct interpretation of the mapping \( (x,y) \to (-x, -y + 5) \) aligns with Option #2: reflect across the x-axis, then up 5.