To solve this problem, we'll follow these steps:

1. Plot the original triangle $\triangle RST$ with vertices $R(4, 1)$, $S(7, 3)$, and $T(6, 4)$.

2. Apply the translation $(x, y) \rightarrow (x-3, y)$ to each vertex.

3. Reflect the translated triangle over the line $y = -1$.

Step 1: Plot the original triangle $\triangle RST$

The vertices of the original triangle are:

• $R(4, 1)$
• $S(7, 3)$
• $T(6, 4)$

Step 2: Apply the translation $(x, y) \rightarrow (x-3, y)$

We subtract 3 from the x-coordinate of each vertex:

• $R'(4-3, 1) = R'(1, 1)$
• $S'(7-3, 3) = S'(4, 3)$
• $T'(6-3, 4) = T'(3, 4)$

The translated triangle $\triangle R'S'T'$ has vertices:

• $R'(1, 1)$
• $S'(4, 3)$
• $T'(3, 4)$

Step 3: Reflect the translated triangle over the line $y = -1$

To reflect a point $(x, y)$ over the line $y = -1$, we use the formula: $y' = -1 - (y + 1) = -2 - y$

Applying this to each vertex of the translated triangle:

• $R''(1, 1)$ reflects to $R''(1, -3)$
• $S''(4, 3)$ reflects to $S''(4, -5)$
• $T''(3, 4)$ reflects to $T''(3, -6)$

The reflected triangle $\triangle R''S''T''$ has vertices:

• $R''(1, -3)$
• $S''(4, -5)$
• $T''(3, -6)$

Graphing the Triangles

1. Original Triangle $\triangle RST$

   • $R(4, 1)$
   • $S(7, 3)$
   • $T(6, 4)$

2. Translated Triangle $\triangle R'S'T'$

   • $R'(1, 1)$
   • $S'(4, 3)$
   • $T'(3, 4)$

3. Reflected Triangle $\triangle R''S''T''$

   • $R''(1, -3)$
   • $S''(4, -5)$
   • $T''(3, -6)$

Would you like a visual representation of the graph, or do you have any questions?

Questions for Further Exploration:

1. How do you determine the line of reflection's equation in different scenarios?
2. What effect does changing the translation vector have on the position of the triangle?
3. Can a reflection occur across a line that is not horizontal or vertical?
4. How would the translation and reflection change if the order of operations were reversed?
5. What happens to the triangle's orientation after reflection across the x-axis?

Tip:

When reflecting over a line $y = c$, the distance from each point to the line before and after the reflection remains the same but is in opposite directions.