Question
Determine the equation for a sine function that has a horizontal compression by a factor of 3 and a reflection across the x-axis.
A. F(x) = -3 sin x
B. F(x) =sin(-3x)
C. F(x)=sin (1/3 x)
D. F(x)=sin (-1/3 x)
A. F(x) = -3 sin x
B. F(x) =sin(-3x)
C. F(x)=sin (1/3 x)
D. F(x)=sin (-1/3 x)
Answers
Answered by
GPT-4o mini
To determine the equation for a sine function that reflects the properties described (a horizontal compression by a factor of 3 and a reflection across the x-axis), we need to understand the effects of these transformations on the sine function.
1. **Horizontal Compression by a Factor of 3**:
- A horizontal compression by a factor of \( k \) is achieved by replacing \( x \) with \( kx \) in the function. For a horizontal compression by a factor of 3, we would replace \( x \) with \( 3x \).
- So, if we start with \( \sin(x) \), after the horizontal compression, it becomes \( \sin(3x) \).
2. **Reflection Across the X-axis**:
- Reflecting a function across the x-axis requires multiplying the entire function by -1. Thus, we take \( \sin(3x) \) and reflect it across the x-axis to get \( -\sin(3x) \).
Combining these transformations gives us the function:
\[
F(x) = -\sin(3x)
\]
None of the options provided directly match \( -\sin(3x) \), but let's rewrite it to check if it resembles any of the given answers.
The choices given are:
- A. \( F(x) = -3 \sin x \)
- B. \( F(x) = \sin(-3x) \)
- C. \( F(x) = \sin \left(\frac{1}{3} x\right) \)
- D. \( F(x) = \sin \left(-\frac{1}{3} x\right) \)
From the transformations analyzed:
- The closest option regarding reflection should involve a negative sign out front of the sine function.
- However, it's important to note that the horizontal compression factor indicated in the provided choices doesn’t correctly reflect our derived compression of \( F(x) = -\sin(3x) \).
Since the options presented do not directly match the derived result of horizontal compression (by 3) and reflection (across the x-axis), we can reaffirm:
None of the provided multiple-choice options is correct for the desired transformations of the sine function based on the requirements you provided.
1. **Horizontal Compression by a Factor of 3**:
- A horizontal compression by a factor of \( k \) is achieved by replacing \( x \) with \( kx \) in the function. For a horizontal compression by a factor of 3, we would replace \( x \) with \( 3x \).
- So, if we start with \( \sin(x) \), after the horizontal compression, it becomes \( \sin(3x) \).
2. **Reflection Across the X-axis**:
- Reflecting a function across the x-axis requires multiplying the entire function by -1. Thus, we take \( \sin(3x) \) and reflect it across the x-axis to get \( -\sin(3x) \).
Combining these transformations gives us the function:
\[
F(x) = -\sin(3x)
\]
None of the options provided directly match \( -\sin(3x) \), but let's rewrite it to check if it resembles any of the given answers.
The choices given are:
- A. \( F(x) = -3 \sin x \)
- B. \( F(x) = \sin(-3x) \)
- C. \( F(x) = \sin \left(\frac{1}{3} x\right) \)
- D. \( F(x) = \sin \left(-\frac{1}{3} x\right) \)
From the transformations analyzed:
- The closest option regarding reflection should involve a negative sign out front of the sine function.
- However, it's important to note that the horizontal compression factor indicated in the provided choices doesn’t correctly reflect our derived compression of \( F(x) = -\sin(3x) \).
Since the options presented do not directly match the derived result of horizontal compression (by 3) and reflection (across the x-axis), we can reaffirm:
None of the provided multiple-choice options is correct for the desired transformations of the sine function based on the requirements you provided.
There are no human answers yet. A form for humans to post answers is coming very soon!