Questions: Graph the following function: (y=frac52 cos (x-5 pi)) Step 1 of 2: Identify the shape of the more basic function that has been shifted, reflected, stretched or compressed.

$Graph the following function: (y=frac52 cos (x-5 pi)) Step 1 of 2: Identify the shape of the more basic function that has been shifted, reflected, stretched or compressed.$

Transcript text: Graph the following function: $y=\frac{5}{2} \cos (x-5 \pi)$ Step 1 of 2: Identify the shape of the more basic function that has been shifted, reflected, stretched or compressed.

Solution

Solution Steps

Step 1: Identify the shape of the more basic function

The given function is $ y = \frac{3}{5} \cos \left( \frac{1}{2} x - \frac{\pi}{4} \right) $. The basic function here is $ y = \cos(x) $.

Step 2: Identify the transformations applied to the basic function

Amplitude Change: The coefficient $\frac{3}{5}$ indicates a vertical compression by a factor of $\frac{3}{5}$.
Horizontal Stretch: The coefficient $\frac{1}{2}$ inside the cosine function indicates a horizontal stretch by a factor of 2.
Phase Shift: The term $-\frac{\pi}{4}$ indicates a phase shift to the right by $\frac{\pi}{4}$.

Step 3: Combine the transformations

Combine the identified transformations to understand the overall effect on the basic cosine function:

The amplitude is reduced to $\frac{3}{5}$.
The period of the function is stretched to $4\pi$ (since the period of $\cos(x)$ is $2\pi$ and it is stretched by a factor of 2).
The graph is shifted to the right by $\frac{\pi}{4}$.

Final Answer

The function $ y = \frac{3}{5} \cos \left( \frac{1}{2} x - \frac{\pi}{4} \right) $ is a vertically compressed cosine function with an amplitude of $\frac{3}{5}$, horizontally stretched by a factor of 2, and shifted to the right by $\frac{\pi}{4}$.

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