Select the correct statements in the passage.

Brian wrote a description of the transformations to the parent sine function that result in this function:
[tex]\[ p(x) = -\frac{1}{4} \sin (x + \pi) - 2 \][/tex]

Which statements in his description are true about function [tex]$ p $[/tex]?

1. To create the graph of function [tex]$ p $[/tex], the graph of the parent function undergoes a phase shift right [tex]$\pi$[/tex] units. Then it is vertically compressed by a factor of [tex]$\frac{1}{4}$[/tex] and reflected over the [tex]$y$[/tex]-axis. Next, it is vertically shifted down 2 units.
2. The frequency of function [tex]$ p $[/tex] is the same as the frequency of the parent function; and the amplitude is 4 times the amplitude of the parent function.

Let's analyze Brian's description of the transformations applied to the parent sine function to derive the function [tex]$ p(x) = -\frac{1}{4} \sin (x + \pi) - 2 $[/tex].

1. Phase Shift:

Brian states that there is a phase shift to the right by [tex]$ \pi $[/tex] units. However, this is incorrect. The term [tex]$ x + \pi $[/tex] actually indicates a phase shift to the left by [tex]$ \pi $[/tex] units (since [tex]$ x + \pi $[/tex] effectively shifts the sine curve to the left).

- Phase Shift: False

2. Vertical Compression:

Brian mentions a vertical compression by a factor of [tex]$ \frac{1}{4} $[/tex]. This is correct. The coefficient [tex]$ \frac{1}{4} $[/tex] in front of the sine function indicates a vertical compression (the amplitude is reduced).

- Vertical Compression: True

3. Reflection:

Brian states that the function is reflected over the [tex]$ y $[/tex]-axis. This is incorrect. The negative sign in front of the fraction [tex]$ -\frac{1}{4} $[/tex] indicates a reflection over the [tex]$ x $[/tex]-axis, not the [tex]$ y $[/tex]-axis.

- Reflection: False

4. Vertical Shift:

Brian says the function is vertically shifted down by 2 units. This is correct. The [tex]$ -2 $[/tex] outside the sine function shows a vertical shift downward by 2 units.

- Vertical Shift: True

5. Frequency:

Brian claims that the frequency of the function [tex]$ p $[/tex] is the same as the frequency of the parent function. This is correct. The argument of the sine function, [tex]$ (x + \pi) $[/tex], does not affect the frequency; it remains [tex]$ 1 $[/tex].

- Frequency: True

6. Amplitude:

Brian asserts that the amplitude is 4 times the amplitude of the parent function. This is incorrect. The amplitude is actually [tex]$ \frac{1}{4} $[/tex] of the parent function’s amplitude due to the coefficient [tex]$ \frac{1}{4} $[/tex].

- Amplitude: False

So, putting it all together with the true statements:

- Phase Shift: False
- Vertical Compression: True
- Reflection: False
- Vertical Shift: True
- Frequency: True
- Amplitude: False

The analyzed correct statements are as follows:
- There is a vertical compression by a factor of [tex]$ \frac{1}{4} $[/tex].
- There is a vertical shift down 2 units.
- The frequency of the function [tex]$ p $[/tex] is the same as the frequency of the parent function.