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Sagot :
To determine which statements about the function [tex]\( h(x) = -3 \cos(2x - \pi) + 4 \)[/tex] are correct, we need to break down the function and analyze the transformations applied to the parent function [tex]\( \cos(x) \)[/tex].
1. Horizontal Compression:
The coefficient of [tex]\( x \)[/tex] inside the cosine function is 2. This indicates a horizontal compression by a factor of [tex]\(\frac{1}{2}\)[/tex] because the function [tex]\( \cos(k \cdot x) \)[/tex] is compressed horizontally by a factor of [tex]\(\frac{1}{k}\)[/tex]. Therefore, the statement "horizontally compressed by a factor of [tex]\(\frac{1}{2}\)[/tex]" is correct.
2. Phase Shift:
The term inside the cosine function is [tex]\( 2x - \pi \)[/tex]. To find the phase shift, we set [tex]\( 2x - \pi = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 2x - \pi = 0 \implies 2x = \pi \implies x = \frac{\pi}{2} \][/tex]
This result means there is a phase shift to the right by [tex]\(\frac{\pi}{2}\)[/tex] units. However, in our transformation notation, this would appear as [tex]\( 2(x - \frac{\pi}{2}) \)[/tex], indicating a shift to the right. Therefore, the statement "undergoes a phase shift left [tex]\(\frac{\pi}{2}\)[/tex] units" is incorrect.
3. Vertical Compression and Reflection:
The coefficient of the cosine function is -3. This indicates a vertical stretch by a factor of 3 and a reflection over the x-axis. Hence, the function is vertically stretched (not compressed) by a factor of 3 and reflected. The negative sign indicates reflection, not compression. Therefore, the statement "vertically compressed by a factor of -3" is misleading. The correct statement should be "vertically stretched by a factor of 3 and reflected over the x-axis".
4. Vertical Shift:
The constant term outside the cosine function is +4, which means the graph is shifted upward by 4 units. Therefore, the statement "vertically shifted up 4 units" is correct.
5. Period of the function:
The period of the parent function [tex]\( \cos(x) \)[/tex] is [tex]\( 2\pi \)[/tex]. With a horizontal compression by a factor of [tex]\( \frac{1}{2} \)[/tex], the period of [tex]\( \cos(2x) \)[/tex] becomes:
[tex]\[ \text{New period} = \frac{2\pi}{2} = \pi \][/tex]
The period is indeed half the period of the parent function. Therefore, the statement "The period of function [tex]\( h \)[/tex] is half the period of the parent function" is correct.
6. Amplitude:
The amplitude of the parent function [tex]\( \cos(x) \)[/tex] is 1. For the given function [tex]\( h(x) \)[/tex], the amplitude is the absolute value of the coefficient of the cosine function, which is |−3| = 3. The statement "it has an amplitude 3 units greater than that of the parent function" is misleading because the amplitude is actually 3 units, not 3 units greater than 1. The correct statement should be "The amplitude is 3".
In summary, the correct statements include:
- "horizontally compressed by a factor of [tex]\(\frac{1}{2}\)[/tex]"
- "vertically shifted up 4 units"
- "The period of function [tex]\( h \)[/tex] is half the period of the parent function"
Therefore, Hillary's correct descriptions of the transformations are:
1. To create the graph of function [tex]\( h \)[/tex], the graph of the parent function is horizontally compressed by a factor of [tex]\(\frac{1}{2}\)[/tex].
2. It is vertically shifted up 4 units.
3. The period of function [tex]\( h \)[/tex] is half the period of the parent function.
Note that the corrections we've identified should be taken into account for a complete and accurate description.
1. Horizontal Compression:
The coefficient of [tex]\( x \)[/tex] inside the cosine function is 2. This indicates a horizontal compression by a factor of [tex]\(\frac{1}{2}\)[/tex] because the function [tex]\( \cos(k \cdot x) \)[/tex] is compressed horizontally by a factor of [tex]\(\frac{1}{k}\)[/tex]. Therefore, the statement "horizontally compressed by a factor of [tex]\(\frac{1}{2}\)[/tex]" is correct.
2. Phase Shift:
The term inside the cosine function is [tex]\( 2x - \pi \)[/tex]. To find the phase shift, we set [tex]\( 2x - \pi = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 2x - \pi = 0 \implies 2x = \pi \implies x = \frac{\pi}{2} \][/tex]
This result means there is a phase shift to the right by [tex]\(\frac{\pi}{2}\)[/tex] units. However, in our transformation notation, this would appear as [tex]\( 2(x - \frac{\pi}{2}) \)[/tex], indicating a shift to the right. Therefore, the statement "undergoes a phase shift left [tex]\(\frac{\pi}{2}\)[/tex] units" is incorrect.
3. Vertical Compression and Reflection:
The coefficient of the cosine function is -3. This indicates a vertical stretch by a factor of 3 and a reflection over the x-axis. Hence, the function is vertically stretched (not compressed) by a factor of 3 and reflected. The negative sign indicates reflection, not compression. Therefore, the statement "vertically compressed by a factor of -3" is misleading. The correct statement should be "vertically stretched by a factor of 3 and reflected over the x-axis".
4. Vertical Shift:
The constant term outside the cosine function is +4, which means the graph is shifted upward by 4 units. Therefore, the statement "vertically shifted up 4 units" is correct.
5. Period of the function:
The period of the parent function [tex]\( \cos(x) \)[/tex] is [tex]\( 2\pi \)[/tex]. With a horizontal compression by a factor of [tex]\( \frac{1}{2} \)[/tex], the period of [tex]\( \cos(2x) \)[/tex] becomes:
[tex]\[ \text{New period} = \frac{2\pi}{2} = \pi \][/tex]
The period is indeed half the period of the parent function. Therefore, the statement "The period of function [tex]\( h \)[/tex] is half the period of the parent function" is correct.
6. Amplitude:
The amplitude of the parent function [tex]\( \cos(x) \)[/tex] is 1. For the given function [tex]\( h(x) \)[/tex], the amplitude is the absolute value of the coefficient of the cosine function, which is |−3| = 3. The statement "it has an amplitude 3 units greater than that of the parent function" is misleading because the amplitude is actually 3 units, not 3 units greater than 1. The correct statement should be "The amplitude is 3".
In summary, the correct statements include:
- "horizontally compressed by a factor of [tex]\(\frac{1}{2}\)[/tex]"
- "vertically shifted up 4 units"
- "The period of function [tex]\( h \)[/tex] is half the period of the parent function"
Therefore, Hillary's correct descriptions of the transformations are:
1. To create the graph of function [tex]\( h \)[/tex], the graph of the parent function is horizontally compressed by a factor of [tex]\(\frac{1}{2}\)[/tex].
2. It is vertically shifted up 4 units.
3. The period of function [tex]\( h \)[/tex] is half the period of the parent function.
Note that the corrections we've identified should be taken into account for a complete and accurate description.
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