Let's break down the problem step by step by analyzing the transformations described.
### Laura's Function:
1. Horizontally Compressed by a Factor of [tex]$\frac{1}{3}$[/tex]:
- The horizontal compression affects the input of the cosine function, resulting in a transformation of [tex]$y = \cos(3x)$[/tex]. Here, the factor 3 inside the cosine function compresses the graph horizontally.
2. Reflected Over the [tex]$x$[/tex]-Axis:
- Reflecting the function over the [tex]$x$[/tex]-axis means multiplying the function by -1. This gives us [tex]$y = -\cos(3x)$[/tex].
So, Laura's final function is:
[tex]\[ g(x) = -\cos(3x) \][/tex]
### Becky's Function:
Becky’s function is given directly as:
[tex]\[ f(x) = 3\cos(x - \pi) \][/tex]
This function does the following:
1. Horizontal Shift:
- The term [tex]$(x - \pi)$[/tex] means a horizontal shift to the right by [tex]$\pi$[/tex] units.
2. Vertical Scaling:
- The coefficient 3 in front of the cosine function scales the graph vertically by a factor of 3.
### Summary:
- Laura's Function: [tex]\( g(x) = -\cos(3x) \)[/tex]
- Becky's Function: [tex]\( f(x) = 3\cos(x - \pi) \)[/tex]
Now we need to determine which graph corresponds to each function.
- Laura's Graph: Since it is horizontally compressed and reflected across the [tex]$x$[/tex]-axis, it should look like a cosine wave that oscillates more quickly (three times as fast) and is flipped upside down.
- Becky's Graph: This graph is a standard cosine wave that has been shifted horizontally by [tex]$\pi$[/tex] units to the right and stretched vertically by a factor of 3.
### Conclusion:
From the given understanding, we can summarize:
- Laura’s graph corresponds to [tex]\( g(x) = -\cos(3x) \)[/tex]
- Becky’s graph corresponds to [tex]\( f(x) = 3\cos(x - \pi) \)[/tex]
Thus the correct choices are:
- Laura for [tex]\( g(x) = -\cos(3x) \)[/tex]
- Becky for [tex]\( f(x) = 3\cos(x - \pi) \)[/tex]
