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Sagot :
To analyze the transformations applied to the parent sine function, let's break down the given function [tex]\( m(x) = -\sin \left(\frac{1}{4} x\right) \)[/tex].
1. Shifted down 1 unit:
The parent function [tex]\( \sin(x) \)[/tex] is vertically shifted up or down by adding or subtracting a constant value outside the sine function. There is no such constant outside the sine function in [tex]\( m(x) = -\sin \left(\frac{1}{4} x\right) \)[/tex]. Therefore, the graph is not shifted down 1 unit.
- Answer: False
2. Reflected over the [tex]\( y \)[/tex]-axis:
Reflection over the [tex]\( y \)[/tex]-axis would occur if there were a negative sign inside the sine function with the [tex]\( x \)[/tex] term, i.e., [tex]\( \sin(-x) \)[/tex]. In this case, there is no negative sign inside the argument of the sine function.
- Answer: False
3. Reflected over the [tex]\( x \)[/tex]-axis:
Reflection over the [tex]\( x \)[/tex]-axis is indicated by a negative sign in front of the entire sine function. In this case, [tex]\( m(x) = -\sin \left(\frac{1}{4} x\right) \)[/tex] includes a negative sign in front of the sine function.
- Answer: True
4. Phase shift of [tex]\(\frac{1}{4}\)[/tex] unit to the right:
A phase shift occurs when there is a horizontal shift applied to the argument of the sine function, such as [tex]\( \sin(x - c) \)[/tex]. Here, there is no horizontal shift term added or subtracted inside the sine function.
- Answer: False
5. Frequency decreases by a factor of [tex]\(\frac{1}{4}\)[/tex]:
The frequency of the sine function is affected by the coefficient of the [tex]\( x \)[/tex] term inside the sine function. In [tex]\( \sin(bx) \)[/tex], if [tex]\( b \)[/tex] is less than 1, the frequency decreases. Here, [tex]\( b = \frac{1}{4} \)[/tex], indicating that the frequency is decreased by a factor of [tex]\( \frac{1}{4} \)[/tex].
- Answer: True
6. Frequency increases by a factor of 4:
The frequency of the sine function increases if the coefficient of the [tex]\( x \)[/tex] term inside the sine function is greater than 1. In this case, [tex]\( b = \frac{1}{4} \)[/tex], so the frequency does not increase; it actually decreases.
- Answer: False
Therefore, the correct statements about the transformations are:
- The graph is reflected over the [tex]\( x \)[/tex]-axis.
- The frequency decreases by a factor of [tex]\(\frac{1}{4}\)[/tex].
Hence, these are the correct answers:
- The graph is reflected over the [tex]\( x \)[/tex]-axis. (True)
- The frequency decreases by a factor of [tex]\(\frac{1}{4}\)[/tex]. (True)
1. Shifted down 1 unit:
The parent function [tex]\( \sin(x) \)[/tex] is vertically shifted up or down by adding or subtracting a constant value outside the sine function. There is no such constant outside the sine function in [tex]\( m(x) = -\sin \left(\frac{1}{4} x\right) \)[/tex]. Therefore, the graph is not shifted down 1 unit.
- Answer: False
2. Reflected over the [tex]\( y \)[/tex]-axis:
Reflection over the [tex]\( y \)[/tex]-axis would occur if there were a negative sign inside the sine function with the [tex]\( x \)[/tex] term, i.e., [tex]\( \sin(-x) \)[/tex]. In this case, there is no negative sign inside the argument of the sine function.
- Answer: False
3. Reflected over the [tex]\( x \)[/tex]-axis:
Reflection over the [tex]\( x \)[/tex]-axis is indicated by a negative sign in front of the entire sine function. In this case, [tex]\( m(x) = -\sin \left(\frac{1}{4} x\right) \)[/tex] includes a negative sign in front of the sine function.
- Answer: True
4. Phase shift of [tex]\(\frac{1}{4}\)[/tex] unit to the right:
A phase shift occurs when there is a horizontal shift applied to the argument of the sine function, such as [tex]\( \sin(x - c) \)[/tex]. Here, there is no horizontal shift term added or subtracted inside the sine function.
- Answer: False
5. Frequency decreases by a factor of [tex]\(\frac{1}{4}\)[/tex]:
The frequency of the sine function is affected by the coefficient of the [tex]\( x \)[/tex] term inside the sine function. In [tex]\( \sin(bx) \)[/tex], if [tex]\( b \)[/tex] is less than 1, the frequency decreases. Here, [tex]\( b = \frac{1}{4} \)[/tex], indicating that the frequency is decreased by a factor of [tex]\( \frac{1}{4} \)[/tex].
- Answer: True
6. Frequency increases by a factor of 4:
The frequency of the sine function increases if the coefficient of the [tex]\( x \)[/tex] term inside the sine function is greater than 1. In this case, [tex]\( b = \frac{1}{4} \)[/tex], so the frequency does not increase; it actually decreases.
- Answer: False
Therefore, the correct statements about the transformations are:
- The graph is reflected over the [tex]\( x \)[/tex]-axis.
- The frequency decreases by a factor of [tex]\(\frac{1}{4}\)[/tex].
Hence, these are the correct answers:
- The graph is reflected over the [tex]\( x \)[/tex]-axis. (True)
- The frequency decreases by a factor of [tex]\(\frac{1}{4}\)[/tex]. (True)
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