To determine the minimum value of the function [tex]\( y = -2 + 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex], you need to analyze the behavior of the sine component within the function.
1. Consider the sine function: [tex]\(\sin \theta\)[/tex]. The sine function, [tex]\(\sin \theta\)[/tex], oscillates between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
2. In the expression [tex]\( 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex], the sine function [tex]\(\sin \left( \frac{\pi}{12} (x - 2) \right)\)[/tex] will also oscillate between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex], regardless of the argument [tex]\(\frac{\pi}{12} (x - 2)\)[/tex].
3. To find the maximum and minimum values of [tex]\(5 \sin \left( \frac{\pi}{12} (x - 2) \right)\)[/tex], we multiply the range [tex]\([-1, 1]\)[/tex] of [tex]\(\sin \theta\)[/tex] by the coefficient [tex]\(5\)[/tex]:
[tex]\[
5 \times -1 = -5 \quad \text{and} \quad 5 \times 1 = 5
\][/tex]
Hence, the expression [tex]\(5 \sin \left( \frac{\pi}{12} (x - 2) \right)\)[/tex] oscillates between [tex]\(-5\)[/tex] and [tex]\(5\)[/tex].
4. Now, consider the full expression [tex]\( y = -2 + 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex]. To find the minimum value of [tex]\(y\)[/tex], we need the smallest value of the term [tex]\( 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex].
- The minimum value of [tex]\( 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex] is [tex]\(-5\)[/tex].
5. Substitute this minimum value back into the expression for [tex]\( y \)[/tex]:
[tex]\[
y = -2 + (-5)
\][/tex]
6. Simplify the expression:
[tex]\[
y = -2 - 5 = -7
\][/tex]
Therefore, the minimum value of the function [tex]\( y = -2 + 5 \sin \left( \frac{\pi}{12} (x - 2) \right) \)[/tex] is [tex]\(\boxed{-7}\)[/tex].
