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Sagot :
Let's break down the transformations step by step based on the function [tex]\( h(x) = -3 \cos (2x - \pi) + 4 \)[/tex]:
### Step 1: Horizontal Compression
The term [tex]\(2x\)[/tex] inside the cosine function indicates a horizontal compression by a factor of [tex]\( \frac{1}{2} \)[/tex].
### Step 2: Phase Shift
The term [tex]\( -\pi \)[/tex] inside the cosine function indicates a phase shift. We solve for the shift by setting the inside equal to 0:
[tex]\[ 2x - \pi = 0 \implies x = \frac{\pi}{2} \][/tex]
This means the graph is shifted to the right by [tex]\( \frac{\pi}{2} \)[/tex] units.
### Step 3: Vertical Compression and Shift
\- The leading coefficient [tex]\( -3 \)[/tex] represents a vertical stretch by a factor of 3 and a reflection across the x-axis.
\- The constant term [tex]\( +4 \)[/tex] represents a vertical shift upward by 4 units.
### Step 4: Period
The period of the parent cosine function is [tex]\( 2\pi \)[/tex]. Given the horizontal compression,
[tex]\[ \text{New Period} = \frac{2\pi}{2} = \pi \][/tex]
### Step 5: Amplitude
The amplitude of the parent cosine function is 1. Given the vertical stretch,
[tex]\[ \text{New Amplitude} = | -3 | = 3 \][/tex]
### Analysis of Statements
Let's analyze the statements provided in Hillary's description:
1. "The graph of the parent function is horizontally compressed by a factor of [tex]\( \frac{1}{\frac{1}{3}} \)[/tex]."
- This statement is incorrect. The horizontal compression factor is actually [tex]\( \frac{1}{2} \)[/tex].
2. "Phase shift left [tex]\( \frac{7}{2} \)[/tex] units."
- This statement is incorrect. The phase shift is to the right by [tex]\( \frac{\pi}{2} \)[/tex] units.
3. "Vertically compressed by a factor of -3 and vertically shifted up 4 units."
- This statement is partially correct regarding the vertical compression (it involves a reflection and stretch by 3). The shift up 4 units part is correct.
4. "The period of function [tex]\( h \)[/tex] is half the period of the parent function."
- This statement is correct. The new period is [tex]\( \pi \)[/tex], which is half of [tex]\( 2\pi \)[/tex].
5. "It has an amplitude 3 units greater than that of the parent function."
- This statement is incorrect. The amplitude is 3, which is not 3 units greater but rather the absolute coefficient of the cosine.
### Correct Statements
The correct and true statements about function [tex]\( h(x) \)[/tex] from Hillary's description would therefore be:
1. It undergoes a horizontal compression by a factor of [tex]\( \frac{1}{2} \)[/tex].
2. It undergoes a phase shift to the right by [tex]\( \frac{\pi}{2} \)[/tex] units.
3. It is vertically stretched by a factor of 3, reflected across the x-axis, and shifted upward by 4 units.
4. The period of the function [tex]\( h \)[/tex] is half the period of the parent function.
5. The amplitude of the function [tex]\( h \)[/tex] is 3.
Thus, only these points should be considered as true descriptions of the transformations.
### Step 1: Horizontal Compression
The term [tex]\(2x\)[/tex] inside the cosine function indicates a horizontal compression by a factor of [tex]\( \frac{1}{2} \)[/tex].
### Step 2: Phase Shift
The term [tex]\( -\pi \)[/tex] inside the cosine function indicates a phase shift. We solve for the shift by setting the inside equal to 0:
[tex]\[ 2x - \pi = 0 \implies x = \frac{\pi}{2} \][/tex]
This means the graph is shifted to the right by [tex]\( \frac{\pi}{2} \)[/tex] units.
### Step 3: Vertical Compression and Shift
\- The leading coefficient [tex]\( -3 \)[/tex] represents a vertical stretch by a factor of 3 and a reflection across the x-axis.
\- The constant term [tex]\( +4 \)[/tex] represents a vertical shift upward by 4 units.
### Step 4: Period
The period of the parent cosine function is [tex]\( 2\pi \)[/tex]. Given the horizontal compression,
[tex]\[ \text{New Period} = \frac{2\pi}{2} = \pi \][/tex]
### Step 5: Amplitude
The amplitude of the parent cosine function is 1. Given the vertical stretch,
[tex]\[ \text{New Amplitude} = | -3 | = 3 \][/tex]
### Analysis of Statements
Let's analyze the statements provided in Hillary's description:
1. "The graph of the parent function is horizontally compressed by a factor of [tex]\( \frac{1}{\frac{1}{3}} \)[/tex]."
- This statement is incorrect. The horizontal compression factor is actually [tex]\( \frac{1}{2} \)[/tex].
2. "Phase shift left [tex]\( \frac{7}{2} \)[/tex] units."
- This statement is incorrect. The phase shift is to the right by [tex]\( \frac{\pi}{2} \)[/tex] units.
3. "Vertically compressed by a factor of -3 and vertically shifted up 4 units."
- This statement is partially correct regarding the vertical compression (it involves a reflection and stretch by 3). The shift up 4 units part is correct.
4. "The period of function [tex]\( h \)[/tex] is half the period of the parent function."
- This statement is correct. The new period is [tex]\( \pi \)[/tex], which is half of [tex]\( 2\pi \)[/tex].
5. "It has an amplitude 3 units greater than that of the parent function."
- This statement is incorrect. The amplitude is 3, which is not 3 units greater but rather the absolute coefficient of the cosine.
### Correct Statements
The correct and true statements about function [tex]\( h(x) \)[/tex] from Hillary's description would therefore be:
1. It undergoes a horizontal compression by a factor of [tex]\( \frac{1}{2} \)[/tex].
2. It undergoes a phase shift to the right by [tex]\( \frac{\pi}{2} \)[/tex] units.
3. It is vertically stretched by a factor of 3, reflected across the x-axis, and shifted upward by 4 units.
4. The period of the function [tex]\( h \)[/tex] is half the period of the parent function.
5. The amplitude of the function [tex]\( h \)[/tex] is 3.
Thus, only these points should be considered as true descriptions of the transformations.
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