To solve the problem of finding the final position of a point after it undergoes two translations, let’s follow a step-by-step process.
1. Identify the initial position of the point:
- The point starts at the position [tex]\((-3, 1)\)[/tex].
2. Apply the first translation:
- The first translation vector is [tex]\(\langle 8, -4 \rangle\)[/tex].
- To apply this translation, add the components of the vector to the initial coordinates of the point.
- For the [tex]\(x\)[/tex]-coordinate: [tex]\(-3 + 8 = 5\)[/tex]
- For the [tex]\(y\)[/tex]-coordinate: [tex]\(1 - 4 = -3\)[/tex]
- After the first translation, the position of the point is [tex]\((5, -3)\)[/tex].
3. Apply the second translation:
- The second translation vector is [tex]\(\langle -2, -2 \rangle\)[/tex].
- To apply this translation, add the components of the vector to the new position of the point.
- For the [tex]\(x\)[/tex]-coordinate: [tex]\(5 - 2 = 3\)[/tex]
- For the [tex]\(y\)[/tex]-coordinate: [tex]\(-3 - 2 = -5\)[/tex]
- After the second translation, the position of the point is [tex]\((3, -5)\)[/tex].
4. Final result:
- The ordered pair for the point’s final position is [tex]\((3, -5)\)[/tex].
So, after undergoing the translations [tex]\(T_{\langle 8,-4\rangle} \text { and } T_{\langle -2,-2\rangle}\)[/tex], the final position of the point, initially at [tex]\((-3, 1)\)[/tex], is [tex]\((3, -5)\)[/tex].
