Question

laura and becky are each graphing a transformation of the parent cosine function. lauras function is a transformation where the parent function is horizontally compressed by a factor of 1/3 and is reflected over the x - axis. beckys function is defined by the equation f(x)=3cos(x - π). determine which graph belongs to each student.

Explanation:

Step1: Analyze Laura's function

Laura's function has only horizontal - compression by a factor of $\frac{1}{3}$. The general form of a horizontally - compressed cosine function is $y = A\cos(Bx)$ where $B> 1$ for compression. Here $B = 3$. The amplitude and vertical position remain the same as the parent function $y=\cos(x)$ (assuming $A = 1$ and no vertical shift).

Step2: Analyze Becky's function

The function $f(x)=3\cos(x - \pi)$ can be rewritten using the cosine difference formula $\cos(A - B)=\cos A\cos B+\sin A\sin B$. So $f(x)=3(\cos x\cos\pi+\sin x\sin\pi)=- 3\cos x$. The amplitude is $|A| = 3$ and it is reflected over the $x$ - axis compared to the parent function $y = \cos x$.

Step3: Match the graphs

The graph with an amplitude of 3 and reflected over the $x$ - axis (starts at $(0, - 3)$) belongs to Becky. The graph with a horizontal compression (more cycles in the same $x$ - interval) and an amplitude of 1 belongs to Laura.

Answer:

Match the graph with amplitude 1 and horizontal compression to Laura. Match the graph with amplitude 3 and reflected over the $x$ - axis (starts at $(0,-3)$) to Becky.