Here are some visual problems that involve translating, reflecting, rotating, and dilating objects across a graph. Each problem is described step-by-step for you to visualize or draw on a graph.

Problem 1: Translation

1. Original Object: Consider a triangle with vertices at $A(1, 2)$, $B(3, 4)$, and $C(2, 6)$.
2. Task: Translate the triangle 4 units to the right and 3 units down.
3. Question: What are the new coordinates of the vertices after the translation?

Problem 2: Reflection

1. Original Object: Consider a square with vertices at $P(-2, 3)$, $Q(1, 3)$, $R(1, 6)$, and $S(-2, 6)$.
2. Task: Reflect the square across the y-axis.
3. Question: What are the new coordinates of the vertices after the reflection?

Problem 3: Rotation

1. Original Object: Consider a line segment with endpoints at $M(2, 1)$ and $N(5, 1)$.
2. Task: Rotate the line segment 90° counterclockwise around the origin.
3. Question: What are the new coordinates of the endpoints after the rotation?

Problem 4: Dilation

1. Original Object: Consider a rectangle with vertices at $J(1, 1)$, $K(4, 1)$, $L(4, 3)$, and $M(1, 3)$.
2. Task: Dilate the rectangle by a scale factor of 2 with the origin as the center of dilation.
3. Question: What are the new coordinates of the vertices after the dilation?

Problem 5: Combination of Transformations

1. Original Object: Consider a parallelogram with vertices at $X(-3, 2)$, $Y(-1, 4)$, $Z(1, 2)$, and $W(-1, 0)$.
2. Task: Reflect the parallelogram across the x-axis, and then translate it 2 units to the left.
3. Question: What are the new coordinates of the vertices after both transformations?

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Would you like detailed solutions for any of these problems or have any questions?

Relative Questions:

1. How do you determine the new coordinates after a translation?
2. What is the effect of reflecting an object across the x-axis versus the y-axis?
3. How does the angle of rotation affect the coordinates of a shape on a graph?
4. What happens to the size and position of a shape when it is dilated by a factor of less than 1?
5. Can you combine more than two transformations, and how would that affect the final result?

Tip: When performing a transformation, always track each vertex of the shape individually to ensure accuracy in your final result.