We are given the transformed function $f(x) = -3h(x - 4) - 2$, and we need to determine the transformations applied to the parent function $h(x)$. We need to identify the reflections, stretches/compressions, vertical shifts, and horizontal shifts.

1. Problem Description

We are given the transformed function $f(x) = -3h(x - 4) - 2$, and we need to determine the transformations applied to the parent function $h(x)$. We need to identify the reflections, stretches/compressions, vertical shifts, and horizontal shifts.

2. Solution Steps

The general transformation of a function $h(x)$ can be represented as $f(x) = a \cdot h(b(x - c)) + d$.

Here, $f(x) = -3h(x-4) - 2$, so we have:

$a = -3$

$b = 1$

$c = 4$

$d = -2$

a) Reflections: The negative sign in front of $3$ (i.e., $a = -3$) indicates a reflection over the x-axis.

b) Stretches/Compressions: The absolute value of $a$ is $|-3| = 3$. Since $|a| > 1$, there is a vertical stretch by a factor of $3$. Since $b = 1$, there is no horizontal stretch or compression.

c) Vertical Shifts: The term $d = -2$ indicates a vertical shift downwards by 2 units.

d) Horizontal Shifts: The term $x - c = x - 4$ indicates a horizontal shift to the right by 4 units.

3. Final Answer

a) Reflection over the x-axis.

b) Vertical Stretch by a factor of

3. c) Vertical Shift down by 2 units.

d) Horizontal Shift right by 4 units.