Sure, let's analyze the transformations of the parent cosine function for Laura and Becky.
### Laura's Function
1. Horizontal Compression by a Factor of [tex]\(\frac{1}{3}\)[/tex]:
- The standard cosine function is [tex]\(\cos(x)\)[/tex].
- A horizontal compression by [tex]\(\frac{1}{3}\)[/tex] means we need to replace [tex]\(x\)[/tex] with [tex]\(3x\)[/tex] in the function. So, [tex]\(\cos(x)\)[/tex] becomes [tex]\(\cos(3x)\)[/tex].
2. Reflection over the [tex]\(x\)[/tex]-Axis:
- Reflecting a function over the [tex]\(x\)[/tex]-axis means multiplying the function by [tex]\(-1\)[/tex].
- Thus, [tex]\(\cos(3x)\)[/tex] becomes [tex]\(-\cos(3x)\)[/tex].
So, Laura's function can be written as:
[tex]\[
f(x) = -\cos(3x)
\][/tex]
### Becky's Function
1. Given Function:
- Becky's function is given directly as [tex]\(f(x) = 3 \cos(x - \pi)\)[/tex].
2. Analysis of Transformations:
- [tex]\(\cos(x - \pi)\)[/tex]: The term [tex]\((x - \pi)\)[/tex] represents a horizontal shift to the right by [tex]\(\pi\)[/tex] units.
- Multiplying [tex]\(\cos(x - \pi)\)[/tex] by 3 stretches the function vertically by a factor of 3.
So, Becky's function can be written as:
[tex]\[
f(x) = 3 \cos(x - \pi)
\][/tex]
### Summary
- Laura's function after transformations is [tex]\(f(x) = -\cos(3x)\)[/tex].
- Becky's function after transformations is [tex]\(f(x) = 3 \cos(x - \pi)\)[/tex].
Each function corresponds to the described transformations, thereby correctly identifying which function graph belongs to each student.
