To determine the maximum value of the function [tex]\( y = -1 + 6 \cos \left( \frac{2\pi}{7} (x - 5) \right) \)[/tex], let's proceed through the following steps:
1. Understand the Cosine Function Range:
The function inside the cosine, [tex]\( \cos \left( \frac{2\pi}{7} (x - 5) \right) \)[/tex], ranges between -1 and 1. This is a property of the cosine function, which oscillates between these extremes regardless of the input.
2. Determine the Maximum Cosine Value:
Since [tex]\( \cos(\theta) \)[/tex] achieves its maximum value of 1, let's substitute this maximum value into the original function to find the corresponding maximum value of [tex]\( y \)[/tex].
3. Substitute the Maximum Cosine Value into the Function:
[tex]\[
y = -1 + 6 \cos \left( \frac{2\pi}{7} (x - 5) \right)
\][/tex]
Substituting 1 for [tex]\( \cos \left( \frac{2\pi}{7} (x - 5) \right) \)[/tex]:
[tex]\[
y = -1 + 6 \cdot 1
\][/tex]
4. Perform the Arithmetic:
Simplifying the equation:
[tex]\[
y = -1 + 6 = 5
\][/tex]
Thus, the maximum value of the function [tex]\( y = -1 + 6 \cos \left( \frac{2\pi}{7} (x - 5) \right) \)[/tex] is [tex]\(\boxed{5}\)[/tex].
