Answer :
Let's analyze the transformations applied to the cosine function [tex]\( y = \cos(x) \)[/tex] to produce [tex]\( y = 3\cos(-2x) \)[/tex] step by step.
1. Horizontal Compression and Reflection:
The function [tex]\(\cos(-2x)\)[/tex] implies two transformations related to the x-variable inside the cosine function:
- The coefficient [tex]\(-2\)[/tex] causes a horizontal compression by a factor of 2. This is because the term [tex]\( -2x \)[/tex] affects the period of the cosine function, which is [tex]\( 2\pi / |2| = \pi \)[/tex], making the function complete one full cycle in half the usual length (compression by factor 2).
- The negative sign before the [tex]\(2\)[/tex] means a reflection across the y-axis. The cosine function is even, meaning [tex]\( \cos(-x) = \cos(x) \)[/tex], so the negative sign affects other functions differently, but it induces a reflection for transformations.
2. Vertical Stretch:
The coefficient [tex]\( 3 \)[/tex] in front of the cosine function, [tex]\( 3\cos(-2x) \)[/tex], implies a vertical stretch by a factor of 3. This transformation multiplies the amplitude of the cosine function by 3, making the peaks and troughs three times as high (or low) as they were originally.
After considering these transformations:
- Horizontal compression by factor 2
- Reflection across the y-axis
- Vertical stretch by factor 3
We can match these steps to one of the given descriptions. Therefore, the correct answer is:
[tex]\[ \text{B. Horizontal compression by factor 2, reflection across the } y\text{-axis, then vertical stretch by factor 3} \][/tex]
1. Horizontal Compression and Reflection:
The function [tex]\(\cos(-2x)\)[/tex] implies two transformations related to the x-variable inside the cosine function:
- The coefficient [tex]\(-2\)[/tex] causes a horizontal compression by a factor of 2. This is because the term [tex]\( -2x \)[/tex] affects the period of the cosine function, which is [tex]\( 2\pi / |2| = \pi \)[/tex], making the function complete one full cycle in half the usual length (compression by factor 2).
- The negative sign before the [tex]\(2\)[/tex] means a reflection across the y-axis. The cosine function is even, meaning [tex]\( \cos(-x) = \cos(x) \)[/tex], so the negative sign affects other functions differently, but it induces a reflection for transformations.
2. Vertical Stretch:
The coefficient [tex]\( 3 \)[/tex] in front of the cosine function, [tex]\( 3\cos(-2x) \)[/tex], implies a vertical stretch by a factor of 3. This transformation multiplies the amplitude of the cosine function by 3, making the peaks and troughs three times as high (or low) as they were originally.
After considering these transformations:
- Horizontal compression by factor 2
- Reflection across the y-axis
- Vertical stretch by factor 3
We can match these steps to one of the given descriptions. Therefore, the correct answer is:
[tex]\[ \text{B. Horizontal compression by factor 2, reflection across the } y\text{-axis, then vertical stretch by factor 3} \][/tex]
