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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.

Sagot :

GinnyAnswer
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.
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