To express a cosine function in a generalized form, we use the following structure:
[tex]\[ y = A \cos \left( \frac{2\pi}{T} (x - P) \right) + V \][/tex]
Here's a detailed explanation of each component in this formula:
1. Amplitude (A):
- This is the coefficient in front of the cosine function, denoted as [tex]\( A \)[/tex].
- It represents the maximum value of the function's displacement from its equilibrium position (the vertical stretch or shrink).
2. Period (T):
- [tex]\( T \)[/tex] is the period of the function, indicating how long it takes for the function to complete one full cycle.
- The term [tex]\(\frac{2\pi}{T}\)[/tex] inside the cosine function adjusts the frequency of the cosine wave accordingly.
3. Phase Shift (P):
- [tex]\( P \)[/tex] represents the horizontal shift along the x-axis.
- If [tex]\( P \)[/tex] is positive, the graph shifts to the right by [tex]\( P \)[/tex] units; if [tex]\( P \)[/tex] is negative, the graph shifts to the left by [tex]\( |P| \)[/tex] units.
4. Vertical Shift (V):
- This is the constant added at the end of the function, denoted as [tex]\( V \)[/tex].
- It represents the upward or downward shift of the entire graph.
Given these components, we can rewrite the given structure to match the definitions:
[tex]\[ y = A \cos \left( \frac{2\pi}{T} (x - P) \right) + V \][/tex]
In this context:
- The amplitude, [tex]\( A \)[/tex], remains a general placeholder.
- The period, represented by [tex]\( T \)[/tex], defines the frequency.
- The phase shift, denoted by [tex]\( P \)[/tex], indicates the horizontal displacement.
- The vertical shift, denoted by [tex]\( V \)[/tex], adjusts the graph vertically.
Therefore, the formula for the function [tex]\( y \)[/tex] in the given structure is:
[tex]\[ y = A \cos \left( \frac{2 \pi}{T} (x - P) \right) + V \][/tex]
