Certainly! Let's break down each transformation step-by-step to understand the changes made to the coordinates:
### Transformation A: [tex]\((x, y) \Rightarrow (x + 3, y = 5)\)[/tex]
1. Translation in the [tex]\( x \)[/tex]-coordinate:
- The [tex]\( x \)[/tex]-coordinate of the point is increased by 3.
- This means the point is shifted 3 units to the right.
2. Fixed [tex]\( y \)[/tex]-coordinate:
- The [tex]\( y \)[/tex]-coordinate is set to 5, regardless of its initial value.
So, any point [tex]\((x, y)\)[/tex] will be transformed to [tex]\((x + 3, 5)\)[/tex].
### Transformation E: [tex]\((2, y) \rightarrow (-5, y + 3)\)[/tex]
1. Fixed [tex]\( x \)[/tex]-coordinate:
- The [tex]\( x \)[/tex]-coordinate is changed to -5, no matter the initial value.
2. Translation in the [tex]\( y \)[/tex]-coordinate:
- The [tex]\( y \)[/tex]-coordinate is increased by 3.
Given that the initial coordinate starts with [tex]\( x = 2 \)[/tex], it doesn't impact the resulting [tex]\( x \)[/tex]-coordinate because it's fixed at -5.
So, any point [tex]\((2, y)\)[/tex] will be transformed to [tex]\((-5, y + 3)\)[/tex].
### Transformation C: [tex]\((x, y) \Rightarrow (x + 5, y = 3)\)[/tex]
1. Translation in the [tex]\( x \)[/tex]-coordinate:
- The [tex]\( x \)[/tex]-coordinate of the point is increased by 5.
- This means the point is shifted 5 units to the right.
2. Fixed [tex]\( y \)[/tex]-coordinate:
- The [tex]\( y \)[/tex]-coordinate is set to 3, regardless of its initial value.
So, any point [tex]\((x, y)\)[/tex] will be transformed to [tex]\((x + 5, 3)\)[/tex].
### Transformation 0: [tex]\((2, 4) \rightarrow (4 - 3, 4 + 8)\)[/tex]
1. Translation in the [tex]\( x \)[/tex]-coordinate:
- The initial [tex]\( x \)[/tex]-coordinate (2) is moved to [tex]\(4 - 3\)[/tex].
- Calculating this, [tex]\(4 - 3\)[/tex] results in 1.
2. Translation in the [tex]\( y \)[/tex]-coordinate:
- The initial [tex]\( y \)[/tex]-coordinate (4) is moved to [tex]\(4 + 8\)[/tex].
- Calculating this, [tex]\(4 + 8\)[/tex] results in 12.
So, the specific point [tex]\((2, 4)\)[/tex] will be transformed to [tex]\((1, 12)\)[/tex].
In summary:
- Transformation A: [tex]\((x, y) \rightarrow (x + 3, 5)\)[/tex]
- Transformation E: [tex]\((2, y) \rightarrow (-5, y + 3)\)[/tex]
- Transformation C: [tex]\((x, y) \rightarrow (x + 5, 3)\)[/tex]
- Transformation 0: [tex]\((2, 4) \rightarrow (1, 12)\)[/tex]
