To determine which graph belongs to Laura and which belongs to Becky based on their respective transformations, let's analyze each function step by step:
### Laura's Function
Laura's transformation involves two key changes to the parent cosine function [tex]\( \cos(x) \)[/tex]:
1. Horizontal Compression by a Factor of [tex]\(\frac{1}{3}\)[/tex]:
- To horizontally compress the parent function [tex]\( \cos(x) \)[/tex] by a factor of [tex]\(\frac{1}{3}\)[/tex], we modify the argument of the cosine function as [tex]\( \cos(3x) \)[/tex]. This transformation compresses the graph horizontally since the period of the cosine function reduces accordingly.
2. Reflection Over the [tex]\( x \)[/tex]-Axis:
- Reflecting the function over the [tex]\( x \)[/tex]-axis involves multiplying the function by [tex]\(-1\)[/tex]. Hence, the function becomes [tex]\( -\cos(3x) \)[/tex].
Combining these transformations, Laura's transformed function is:
[tex]\[ f(x) = -\cos(3x) \][/tex]
### Becky's Function
Becky’s function is given explicitly as:
[tex]\[ f(x) = 3 \cos(x - \pi) \][/tex]
To understand this function, let's break down the transformations:
1. Horizontal Translation:
- The term [tex]\((x - \pi)\)[/tex] indicates a horizontal shift of the function to the right by [tex]\(\pi\)[/tex] units.
2. Vertical Stretch:
- The coefficient [tex]\(3\)[/tex] in front of the cosine function stretches the graph vertically by a factor of [tex]\(3\)[/tex].
Combining these, Becky's function remains:
[tex]\[ f(x) = 3 \cos(x - \pi) \][/tex]
### Summary
- Laura's function: [tex]\( f(x) = -\cos(3x) \)[/tex]
- Becky's function: [tex]\( f(x) = 3 \cos(x - \pi) \)[/tex]
Using the above analysis, we can match each student to their graph transformations:
- Laura's function is [tex]\( -\cos(3x) \)[/tex]
- Becky's function is [tex]\( 3 \cos(x - \pi) \)[/tex]
Therefore, the correct assignments for each student's function description are:
- For Laura: [tex]\(-\cos(3x)\)[/tex]
- For Becky: [tex]\(3 \cos(x - \pi)\)[/tex]
