Sure! Let's work through this together step-by-step.
First, we are given the coordinates of point [tex]\( G \)[/tex] which are [tex]\((-4, 6)\)[/tex]. We are also told that this point is moved horizontally 4 units to the left.
### Understanding Horizontal Translation
1. Horizontal Translation: Moving a point horizontally affects its x-coordinate while the y-coordinate remains unchanged. When we move a point to the left, we subtract from the x-coordinate.
Given these details:
- The initial coordinates of the point [tex]\( G \)[/tex] are [tex]\((-4, 6)\)[/tex].
- To move horizontally to the left by 4 units, we subtract 4 from the x-coordinate.
Let's apply the translation:
[tex]\[ x' = -4 - 4 \][/tex]
[tex]\[ y' = 6 \][/tex]
Simplifying this:
[tex]\[ x' = -8 \][/tex]
[tex]\[ y' = 6 \][/tex]
So, the new coordinates of point [tex]\( G \)[/tex] after the translation are [tex]\((-8, 6)\)[/tex].
### Determining the Correct Mapping
Given the choices:
1. [tex]\((x, y) \rightarrow (x, y+4)\)[/tex]
2. [tex]\((x, y) \rightarrow (x+4, y)\)[/tex]
3. [tex]\((x, y) \rightarrow (x, y-4)\)[/tex]
4. [tex]\((x, y) \rightarrow (x-4, y)\)[/tex]
We need to match our translation to one of these mappings.
- Option 1: [tex]\((x, y) \rightarrow (x, y+4)\)[/tex]
- This implies moving vertically 4 units up.
- This is not our translation since we moved horizontally.
- Option 2: [tex]\((x, y) \rightarrow (x+4, y)\)[/tex]
- This implies moving horizontally 4 units to the right.
- This is incorrect since we moved to the left.
- Option 3: [tex]\((x, y) \rightarrow (x, y-4)\)[/tex]
- This implies moving vertically 4 units down.
- This is not correct as we are not moving vertically.
- Option 4: [tex]\((x, y) \rightarrow (x-4, y)\)[/tex]
- This implies moving horizontally 4 units to the left.
- This is correct because this matches our translation.
Thus, the correct mapping for moving point [tex]\( G (-4, 6) \)[/tex] 4 units to the left is:
[tex]\[ (x, y) \rightarrow (x-4, y) \][/tex]
The resulting coordinates are [tex]\( (-8, 6) \)[/tex].
Therefore, the mapping [tex]\((x, y) \rightarrow (x-4, y)\)[/tex] results in the coordinates [tex]\((-8, 6)\)[/tex] after the translation 4 units to the left.
