Answer :
To determine which rule describes a translation that is 8 units to the right and 2 units up, let's analyze the effect of such translations on the coordinate [tex]\((x, y)\)[/tex].
### Step-by-Step Analysis:
1. Translation 8 units to the right:
- Moving a point 8 units to the right involves increasing the [tex]\(x\)[/tex]-coordinate by 8. Therefore, the new [tex]\(x\)[/tex]-coordinate will be [tex]\(x + 8\)[/tex].
2. Translation 2 units up:
- Moving a point 2 units up involves increasing the [tex]\(y\)[/tex]-coordinate by 2. Therefore, the new [tex]\(y\)[/tex]-coordinate will be [tex]\(y + 2\)[/tex].
Combining these two transformations, the new coordinates of the point after applying the translation will be:
[tex]\[ (x, y) \rightarrow (x + 8, y + 2) \][/tex]
### Selecting the Correct Rule:
Given the choices:
1. [tex]\((x, y) \rightarrow (x - 8, y + 2)\)[/tex]
2. [tex]\((x, y) \rightarrow (x + 8, y + 2)\)[/tex]
3. [tex]\((x, y) \rightarrow (x + 8, y - 2)\)[/tex]
4. [tex]\((x, y) \rightarrow (x - 8, y - 2)\)[/tex]
By analyzing each rule, we observe:
- Option [tex]\((x, y) \rightarrow (x - 8, y + 2)\)[/tex] translates the point 8 units to the left and 2 units up.
- Option [tex]\((x, y) \rightarrow (x + 8, y + 2)\)[/tex] translates the point 8 units to the right and 2 units up.
- Option [tex]\((x, y) \rightarrow (x + 8, y - 2)\)[/tex] translates the point 8 units to the right and 2 units down.
- Option [tex]\((x, y) \rightarrow (x - 8, y - 2)\)[/tex] translates the point 8 units to the left and 2 units down.
Thus, the rule that correctly describes a translation 8 units to the right and 2 units up is:
[tex]\[ (x, y) \rightarrow (x + 8, y + 2) \][/tex]
### Conclusion:
The correct answer is option 2.
[tex]\[ \boxed{2} \][/tex]
### Step-by-Step Analysis:
1. Translation 8 units to the right:
- Moving a point 8 units to the right involves increasing the [tex]\(x\)[/tex]-coordinate by 8. Therefore, the new [tex]\(x\)[/tex]-coordinate will be [tex]\(x + 8\)[/tex].
2. Translation 2 units up:
- Moving a point 2 units up involves increasing the [tex]\(y\)[/tex]-coordinate by 2. Therefore, the new [tex]\(y\)[/tex]-coordinate will be [tex]\(y + 2\)[/tex].
Combining these two transformations, the new coordinates of the point after applying the translation will be:
[tex]\[ (x, y) \rightarrow (x + 8, y + 2) \][/tex]
### Selecting the Correct Rule:
Given the choices:
1. [tex]\((x, y) \rightarrow (x - 8, y + 2)\)[/tex]
2. [tex]\((x, y) \rightarrow (x + 8, y + 2)\)[/tex]
3. [tex]\((x, y) \rightarrow (x + 8, y - 2)\)[/tex]
4. [tex]\((x, y) \rightarrow (x - 8, y - 2)\)[/tex]
By analyzing each rule, we observe:
- Option [tex]\((x, y) \rightarrow (x - 8, y + 2)\)[/tex] translates the point 8 units to the left and 2 units up.
- Option [tex]\((x, y) \rightarrow (x + 8, y + 2)\)[/tex] translates the point 8 units to the right and 2 units up.
- Option [tex]\((x, y) \rightarrow (x + 8, y - 2)\)[/tex] translates the point 8 units to the right and 2 units down.
- Option [tex]\((x, y) \rightarrow (x - 8, y - 2)\)[/tex] translates the point 8 units to the left and 2 units down.
Thus, the rule that correctly describes a translation 8 units to the right and 2 units up is:
[tex]\[ (x, y) \rightarrow (x + 8, y + 2) \][/tex]
### Conclusion:
The correct answer is option 2.
[tex]\[ \boxed{2} \][/tex]
