Answered

Get personalized and accurate responses to your questions with IDNLearn.com. Discover the information you need from our experienced professionals who provide accurate and reliable answers to all your questions.

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 :

GinnyAnsweridn
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.
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com is committed to providing the best answers. Thank you for visiting, and see you next time for more solutions.