select the correct answer from each drop - down menu. laura and becky are each graphing a transformation of…

select the correct answer from each drop - down menu. 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 1/2 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.

select the correct answer from each drop - down menu. 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 1/2 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 $\frac{1}{2}$ gives $y=\cos(2x)$, and a reflection over the $x$ - axis gives $y =-\cos(2x)$. The period of $y = \cos(x)$ is $2\pi$, and for $y = \cos(2x)$ the period is $\frac{2\pi}{2}=\pi$. The amplitude of $y =-\cos(2x)$ is $1$.

Step2: Analyze Becky's function

Becky's function is $f(x)=3\cos(x - \pi)$. Using the identity $\cos(A - B)=\cos A\cos B+\sin A\sin B$, $\cos(x-\pi)=-\cos x$, so $f(x)=- 3\cos(x)$. The period of $y = \cos(x)$ is $2\pi$, and the amplitude is $3$.

Step3: Match the graphs

The graph with a period of $\pi$ and amplitude $1$ belongs to Laura. The graph with a period of $2\pi$ and amplitude $3$ belongs to Becky.

Answer:

The graph with period $\pi$ and amplitude $1$: Laura; The graph with period $2\pi$ and amplitude $3$: Becky.