To solve this, let's go through the steps one by one:

Step 1: Identify the coordinates of points A, B, and C

From the graph:

• $A = (1, 5)$
• $B = (4, 3)$
• $C = (3, 7)$

Step 2: Reflect over the y-axis

Reflecting each point over the y-axis involves changing the sign of the x-coordinate while keeping the y-coordinate the same.

• $A' = (-1, 5)$
• $B' = (-4, 3)$
• $C' = (-3, 7)$

Thus, the coordinates of $\Delta A'B'C'$ after reflection are:

• $A' = (-1, 5)$
• $B' = (-4, 3)$
• $C' = (-3, 7)$

Step 3: Apply the translation $T(2, -8)$

Now we apply the translation $T(2, -8)$, which means adding 2 to the x-coordinates and subtracting 8 from the y-coordinates of each point.

1. For $A' = (-1, 5)$:

   • $A'' = (-1 + 2, 5 - 8) = (1, -3)$

2. For $B' = (-4, 3)$:

   • $B'' = (-4 + 2, 3 - 8) = (-2, -5)$

3. For $C' = (-3, 7)$:

   • $C'' = (-3 + 2, 7 - 8) = (-1, -1)$

The coordinates of $\Delta A''B''C''$ after both transformations are:

• $A'' = (1, -3)$
• $B'' = (-2, -5)$
• $C'' = (-1, -1)$

Summary:

• Coordinates of $\Delta A'B'C'$ (after reflection):
  $A' = (-1, 5)$, $B' = (-4, 3)$, $C' = (-3, 7)$
• Coordinates of $\Delta A''B''C''$ (after translation):
  $A'' = (1, -3)$, $B'' = (-2, -5)$, $C'' = (-1, -1)$

Would you like further explanation on any of these steps?

Additional Questions to Explore:

1. What is the effect of reflecting a shape over the x-axis instead of the y-axis?
2. How would the coordinates change if the translation rule was $T(-2, 8)$ instead?
3. How do you find the distance between two points on a coordinate plane?
4. What would happen if we first translated and then reflected the triangle?
5. How can you determine the area of $\Delta A''B''C''$?

Tip:

When performing transformations, remember to apply each step systematically—starting with reflections or rotations and following up with translations if they are sequential.