Question

lauras function is a transformation where the parent function is horizontally compressed by a factor of \\(\frac{1}{3}\\) and is reflected over the x-axis. beckys function is defined by the equation \\(f(x) = 3\cos(x - \pi)\\). determine which graph belongs to each student.

Explanation:

Step1: Analyze Becky's function

Start with parent $y=\cos(x)$. Apply phase shift: $f(x)=3\cos(x-\pi) = -3\cos(x)$ (using $\cos(x-\pi)=-\cos x$). This is a reflection over x-axis, vertical stretch by 3, period $2\pi$, amplitude 3.

Step2: Analyze Laura's transformation

Parent $y=\cos(x)$. Horizontal compression by $\frac{1}{3}$: $y=\cos(3x)$. Reflect over x-axis: $y=-\cos(3x)$. Period is $\frac{2\pi}{3}$, amplitude 1.

Step3: Match to graphs

1. Left graph: amplitude 3, period $2\pi$, matches $f(x)=-3\cos(x)$ (Becky's function).
2. Middle graph: amplitude 1, period $2\pi$, matches $y=\cos(x)$ (not either student).
3. Right graph: amplitude 1, period $\frac{2\pi}{3}$, matches Laura's function.

Answer:

• Left Graph: Becky's function ($f(x) = 3\cos(x - \pi)$)
• Right Graph: Laura's transformed function ($y=-\cos(3x)$)
• Middle Graph: Neither student's function