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Sagot :
Certainly! Let's analyze each student's function based on the given transformations and equations.
Laura's Function:
1. Horizontal Compression by a Factor of [tex]\(\frac{1}{3}\)[/tex]:
- The general formula for a horizontal compression by a factor of [tex]\(\frac{1}{3}\)[/tex] of the cosine function [tex]\( \cos(x) \)[/tex] is given by [tex]\( \cos(3x) \)[/tex]. This is because a horizontal compression by a factor of [tex]\(\frac{1}{3}\)[/tex] means the function completes one cycle three times faster than the parent function.
2. Reflection Over the [tex]\( x \)[/tex]-Axis:
- Reflecting over the [tex]\( x \)[/tex]-axis involves multiplying the function by [tex]\(-1\)[/tex]. Therefore, the reflection of [tex]\( \cos(3x) \)[/tex] is [tex]\( -\cos(3x) \)[/tex].
Combining these two transformations, Laura’s function is [tex]\( -\cos(3x) \)[/tex].
Becky's Function:
Given the equation [tex]\( f(x) = 3\cos(x - \pi) \)[/tex]:
1. Phase Shift:
- The term [tex]\( (x - \pi) \)[/tex] indicates a horizontal shift (or phase shift) to the right by [tex]\(\pi\)[/tex] units. This means the whole cosine wave moves [tex]\(\pi\)[/tex] units to the right on the [tex]\( x \)[/tex]-axis.
2. Vertical Scaling:
- The coefficient [tex]\( 3 \)[/tex] represents a vertical stretch. It means that the amplitude of the cosine function is multiplied by 3, making the highest peak 3 units and the lowest trough -3 units.
Putting it all together, Becky’s function is [tex]\( 3\cos(x - \pi) \)[/tex].
Matching to Graphs:
Given these expressions for the transformations:
1. Laura’s Graph: Should show compressions such that the period is three times shorter and also reflected over the [tex]\( x \)[/tex]-axis, effectively flipping the cosine wave upside down.
2. Becky’s Graph: Should appear similar to a standard cosine wave but shifted to the right by [tex]\(\pi\)[/tex] units and with peaks at [tex]\( 3 \)[/tex] and troughs at [tex]\( -3 \)[/tex].
By identifying these characteristics, you can determine which graph corresponds to each student visually.
Laura's Function:
1. Horizontal Compression by a Factor of [tex]\(\frac{1}{3}\)[/tex]:
- The general formula for a horizontal compression by a factor of [tex]\(\frac{1}{3}\)[/tex] of the cosine function [tex]\( \cos(x) \)[/tex] is given by [tex]\( \cos(3x) \)[/tex]. This is because a horizontal compression by a factor of [tex]\(\frac{1}{3}\)[/tex] means the function completes one cycle three times faster than the parent function.
2. Reflection Over the [tex]\( x \)[/tex]-Axis:
- Reflecting over the [tex]\( x \)[/tex]-axis involves multiplying the function by [tex]\(-1\)[/tex]. Therefore, the reflection of [tex]\( \cos(3x) \)[/tex] is [tex]\( -\cos(3x) \)[/tex].
Combining these two transformations, Laura’s function is [tex]\( -\cos(3x) \)[/tex].
Becky's Function:
Given the equation [tex]\( f(x) = 3\cos(x - \pi) \)[/tex]:
1. Phase Shift:
- The term [tex]\( (x - \pi) \)[/tex] indicates a horizontal shift (or phase shift) to the right by [tex]\(\pi\)[/tex] units. This means the whole cosine wave moves [tex]\(\pi\)[/tex] units to the right on the [tex]\( x \)[/tex]-axis.
2. Vertical Scaling:
- The coefficient [tex]\( 3 \)[/tex] represents a vertical stretch. It means that the amplitude of the cosine function is multiplied by 3, making the highest peak 3 units and the lowest trough -3 units.
Putting it all together, Becky’s function is [tex]\( 3\cos(x - \pi) \)[/tex].
Matching to Graphs:
Given these expressions for the transformations:
1. Laura’s Graph: Should show compressions such that the period is three times shorter and also reflected over the [tex]\( x \)[/tex]-axis, effectively flipping the cosine wave upside down.
2. Becky’s Graph: Should appear similar to a standard cosine wave but shifted to the right by [tex]\(\pi\)[/tex] units and with peaks at [tex]\( 3 \)[/tex] and troughs at [tex]\( -3 \)[/tex].
By identifying these characteristics, you can determine which graph corresponds to each student visually.
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