Let's work through the transformations step by step.

Given:

• Triangle $ABC$ with vertices:
  $A(-8, -6), B(-3, -6), C(-2, -3)$.

(a) Reflection over the $y$-axis

For a reflection over the $y$-axis, the rule is:
$(x, y) \to (-x, y)$
Applying this to each vertex:

1. $A(-8, -6) \to A' (8, -6)$
2. $B(-3, -6) \to B' (3, -6)$
3. $C(-2, -3) \to C' (2, -3)$

Coordinates of $\triangle A'B'C'$:

• $A'(8, -6), B'(3, -6), C'(2, -3)$.

(b) Rotation of $180^\circ$ about the origin

The rule for $180^\circ$ rotation about the origin is:
$(x, y) \to (-x, -y)$
Applying this to $A'B'C'$:

1. $A'(8, -6) \to A'' (-8, 6)$
2. $B'(3, -6) \to B'' (-3, 6)$
3. $C'(2, -3) \to C'' (-2, 3)$

Coordinates of $\triangle A''B''C''$:

• $A''(-8, 6), B''(-3, 6), C''(-2, 3)$.

(c) Translation under the rule $(x, y) \to (x, y+4)$

This rule shifts each vertex 4 units up.
Applying this to $A''B''C''$:

1. $A''(-8, 6) \to A''' (-8, 10)$
2. $B''(-3, 6) \to B''' (-3, 10)$
3. $C''(-2, 3) \to C''' (-2, 7)$

Coordinates of $\triangle A'''B'''C'''$:

• $A'''(-8, 10), B'''(-3, 10), C'''(-2, 7)$.

(d) Single transformation to map $\triangle ABC \to \triangle A'''B'''C'''$

From the transformations:

1. Reflection over the $y$-axis
2. $180^\circ$ rotation
3. Translation up by 4 units

These three steps suggest the single transformation is a reflection combined with vertical translation, which ultimately maps $\triangle ABC$ to $\triangle A'''B'''C'''$.

Final Summary:

1. (a): $A'(8, -6), B'(3, -6), C'(2, -3)$
2. (b): $A''(-8, 6), B''(-3, 6), C''(-2, 3)$
3. (c): $A'''(-8, 10), B'''(-3, 10), C'''(-2, 7)$
4. (d): The single transformation combines a reflection and translation.

Questions for Further Practice:

1. What are the rules for reflections over the $x$-axis and origin?
2. How would a $90^\circ$ rotation about the origin affect the coordinates?
3. Can you determine the distance between $A$ and $A'''$ after all transformations?
4. What happens if we reflect $\triangle ABC$ across $y = x$?
5. How do you describe the single transformation algebraically?

Tip: Always apply transformations step by step and verify using coordinate rules.