The problem in the image involves mapping quadrilateral $ABCD$ onto $A'B'C'D'$ using sequences of transformations such as translations, reflections, and rotations.

Explanation of each option:

1. Reflection across the x-axis followed by a translation left 8 units and up 2 units:

   • A reflection across the x-axis flips the figure over the x-axis.
   • After this, translating left by 8 units and up by 2 units seems to correctly move $ABCD$ onto $A'B'C'D'$.
   • This sequence is valid.

2. Translation left 3 units and down 2 units followed by a 180° rotation around the origin:

   • Translating left 3 units and down 2 units moves the figure, and a subsequent 180° rotation around the origin flips and relocates it to match $A'B'C'D'$.
   • This sequence is also valid.

3. Reflection across the y-axis followed by a translation right 3 units and up 10 units:

   • A reflection across the y-axis reverses the orientation over the y-axis.
   • After the reflection, translating right 3 units and up 10 units properly moves $ABCD$ onto $A'B'C'D'$.
   • This sequence is valid as well.

4. Rotation 180° around the origin followed by a translation right 3 units and up 2 units:

   • A 180° rotation around the origin flips $ABCD$ upside down.
   • Translating right 3 units and up 2 units afterward moves the figure to the final position of $A'B'C'D'$.
   • This sequence is valid.

Conclusion:

All four selected options are correct sequences that map quadrilateral $ABCD$ onto $A'B'C'D'$.

Would you like further clarification on any of the transformations?

Related Questions:

1. How do you perform a reflection across the x-axis?
2. What is the effect of a 180° rotation around the origin?
3. How can you determine the correct translation after a reflection?
4. What are the differences between reflecting over the x-axis and y-axis?
5. How does the orientation of the figure change after a 180° rotation?

Tip:

When working with transformations, sketching the steps one by one can help visualize how each movement affects the shape's position.