The problem asks to identify the type of transformation applied to the parent function $g(x)$ to obtain the function $f(x) = g(x) + 9$. The options are reflections, stretches/compressions, vertical shifts, and horizontal shifts.

1. Problem Description

The problem asks to identify the type of transformation applied to the parent function $g(x)$ to obtain the function $f(x) = g(x) + 9$. The options are reflections, stretches/compressions, vertical shifts, and horizontal shifts.

2. Solution Steps

The transformation is given by the equation $f(x) = g(x) + 9$.

This transformation adds a constant value (9) to the output of the function $g(x)$.

Adding a constant to the output of a function results in a vertical shift. If the constant is positive, the shift is upwards. If the constant is negative, the shift is downwards.

In this case, since we are adding 9 to $g(x)$, the graph of $g(x)$ is shifted upwards by 9 units.

Vertical shifts are described by transformations of the form $f(x) = g(x) + k$, where $k$ is a constant. If $k > 0$, the graph shifts upwards by $k$ units. If $k < 0$, the graph shifts downwards by $|k|$ units.

Reflections across the x-axis are described by $f(x) = -g(x)$.

Reflections across the y-axis are described by $f(x) = g(-x)$.

Horizontal stretches/compressions are described by $f(x) = g(cx)$ for some constant $c$.

Horizontal shifts are described by $f(x) = g(x-h)$, where $h$ is a constant.

Since we have $f(x) = g(x) + 9$, this is a vertical shift of the parent function $g(x)$.

3. Final Answer

The transformation is a vertical shift.

Specifically, it is a vertical shift upwards by 9 units.

The answers for each option are:

a) Reflections: No.

b) Stretches/Compressions: No.

c) Vertical Shifts: Yes, upwards by

9. d) Horizontal Shifts: No.