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.
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Solution

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Solution Steps

Step 1: Identify the shape of the more basic function

The given function is y=35cos(12xπ4) y = \frac{3}{5} \cos \left( \frac{1}{2} x - \frac{\pi}{4} \right) . The basic function here is y=cos(x) y = \cos(x) .

Step 2: Identify the transformations applied to the basic function
  • Amplitude Change: The coefficient 35\frac{3}{5} indicates a vertical compression by a factor of 35\frac{3}{5}.
  • Horizontal Stretch: The coefficient 12\frac{1}{2} inside the cosine function indicates a horizontal stretch by a factor of 2.
  • Phase Shift: The term π4-\frac{\pi}{4} indicates a phase shift to the right by π4\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 35\frac{3}{5}.
  • The period of the function is stretched to 4π4\pi (since the period of cos(x)\cos(x) is 2π2\pi and it is stretched by a factor of 2).
  • The graph is shifted to the right by π4\frac{\pi}{4}.

Final Answer

The function y=35cos(12xπ4) y = \frac{3}{5} \cos \left( \frac{1}{2} x - \frac{\pi}{4} \right) is a vertically compressed cosine function with an amplitude of 35\frac{3}{5}, horizontally stretched by a factor of 2, and shifted to the right by π4\frac{\pi}{4}.

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