laura and becky are each graphing a transformation of the parent cosine function. lauras function is a…

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.

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.

Answer

Explanation:

Step1: Find Laura's function

The parent - cosine function is $y = \cos(x)$. A horizontal compression by a factor of $\frac{1}{3}$ gives $y=\cos(3x)$, and a reflection over the $x$ - axis gives $y =-\cos(3x)$. The period of $y =-\cos(3x)$ is $T=\frac{2\pi}{3}$ (using the formula $T=\frac{2\pi}{|b|}$ for $y = A\cos(bx - c)+d$, here $b = 3$).

Step2: Analyze Becky's function

For the function $y = 3\cos(x-\pi)$, we use the identity $\cos(A - B)=\cos A\cos B+\sin A\sin B$. So, $y = 3\cos(x-\pi)=3(\cos x\cos\pi+\sin x\sin\pi)=- 3\cos x$. The period of $y=-3\cos x$ is $T = 2\pi$ (since for $y = A\cos(bx - c)+d$, $b = 1$).

Step3: Match the graphs

The graph with a period of $2\pi$ and amplitude of 3 is for Becky's function $y=-3\cos x$. The graph with a period of $\frac{2\pi}{3}$ and amplitude of 1 is for Laura's function $y =-\cos(3x)$.

Answer:

The graph with period $2\pi$ and amplitude 3 (the first graph in the middle) belongs to Becky. The graph with period $\frac{2\pi}{3}$ and amplitude 1 (the third graph on the right) belongs to Laura.