To determine the maximum value of the function [tex]\( y = -3 + 4 \cos \left( \frac{5 \pi}{6} (x + 4) \right) \)[/tex], we need to consider the properties of the cosine function.
1. Understand the properties of the cosine function:
- The cosine function, [tex]\(\cos(\theta)\)[/tex], ranges from [tex]\(-1\)[/tex] to [tex]\(1\)[/tex].
- This means the output of [tex]\(\cos(\theta)\)[/tex] can be anywhere within this interval: [tex]\(-1 \leq \cos(\theta) \leq 1\)[/tex].
2. Applying this to the given function:
- The function given is [tex]\( y = -3 + 4 \cos \left( \frac{5 \pi}{6} (x + 4) \right) \)[/tex].
- To find the maximum value of this function, we need to determine the maximum value that [tex]\(4 \cos\left(\frac{5 \pi}{6} (x + 4)\right)\)[/tex] can take.
3. Determine the maximum value of the cosine component:
- The maximum value of [tex]\(\cos(\theta)\)[/tex] is [tex]\(1\)[/tex].
- Therefore, the maximum value of [tex]\(4 \cos\left(\frac{5 \pi}{6} (x + 4)\right)\)[/tex] will be [tex]\(4 \cdot 1 = 4\)[/tex].
4. Calculate the maximum value of the given function:
- Substitute the maximum value of the cosine term into the function [tex]\( y = -3 + 4 \cos \left( \frac{5 \pi}{6} (x + 4) \right) \)[/tex].
- Therefore, when [tex]\(\cos\left(\frac{5 \pi}{6} (x + 4)\right) = 1\)[/tex], the function becomes:
[tex]\[
y = -3 + 4 \cdot 1 = -3 + 4 = 1
\][/tex]
Hence, the maximum value of the function [tex]\( y = -3 + 4 \cos \left( \frac{5 \pi}{6} (x + 4) \right) \)[/tex] is [tex]\( \boxed{1} \)[/tex].
