Sure, let's determine the transformations for each student's function step-by-step to identify their graphs.
### Step-by-Step Solution:
1. Laura's Function:
- Horizontal Compression by a Factor of [tex]\( \frac{1}{3} \)[/tex]:
The parent cosine function is [tex]\( \cos(x) \)[/tex]. When it is horizontally compressed by a factor of [tex]\( \frac{1}{3} \)[/tex], the new function will be [tex]\( \cos(3x) \)[/tex]. This is because compressing horizontally by [tex]\( \frac{1}{3} \)[/tex] means multiplying the input by 3.
- Reflection Over the [tex]\( x \)[/tex]-Axis:
Reflecting a function over the [tex]\( x \)[/tex]-axis changes the sign of the function. Therefore, [tex]\( \cos(3x) \)[/tex] will become [tex]\( -\cos(3x) \)[/tex].
So, Laura's transformed function is [tex]\( y = -\cos(3x) \)[/tex].
2. Becky's Function:
- Equation Given:
Becky’s function is defined by the equation [tex]\( f(x) = 3 \cos (x - \pi) \)[/tex].
- Transformation Details:
The given equation includes a cosine function shifted horizontally and scaled vertically:
- Horizontal Shift: The term [tex]\( (x - \pi) \)[/tex] represents a shift to the right by [tex]\( \pi \)[/tex] units.
- Vertical Scaling: The coefficient 3 in front of the cosine function scales the amplitude by 3.
So, Becky's transformed function is [tex]\( y = 3\cos(x - \pi) \)[/tex].
By understanding these transformations, we can match each function to its respective graph:
Laura’s function is [tex]\( y = -\cos(3x) \)[/tex].
Becky’s function is [tex]\( y = 3\cos(x - \pi) \)[/tex].
